Research ArticleOption Price Decomposition in Spot-Dependent VolatilityModels and Some Applications

Rauacutel Merino12 and Josep Vives1

1Universitat de Barcelona Departament de Matematiques i Informatica Gran Via 585 08007 Barcelona Spain2VidaCaixa SA Investment Control Department Juan Gris 2-8 08014 Barcelona Spain

Correspondence should be addressed to Raul Merino raulmerino85gmailcom

Received 3 March 2017 Accepted 11 June 2017 Published 31 July 2017

Academic Editor Henri Schurz

Copyright copy 2017 Raul Merino and Josep VivesThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We obtain a Hull and White type option price decomposition for a general local volatility model We apply the obtained formulato CEV model As an application we give an approximated closed formula for the call option price under a CEV model and anapproximated short term implied volatility surfaceThese approximated formulas are used to estimatemodel parametersNumericalcomparison is performed for our new method with exact and approximated formulas existing in the literature

1 Introduction

In [1] a decomposition of the price of a plain vanilla callunder the Heston model is obtained using Ito calculusRecently in [2] the decomposition obtained in [1] has beenused to infer a closed form approximation formula for a plainvanilla call price in the Heston case and on the basis of thisapproximated price a method to calibrate model parametershas been developed and successfully applied In this paperwe use the ideas presented in [1] to obtain a closed formapproximation to plain vanilla call option price under a spot-dependent volatility model

The model presented here assumes the volatility is adeterministic function of the underlying stock price andtherefore there is only one source of randomness in themodel These models are sometimes called local volatilitymodels in the industry and GARCH-type volatility models infinancial econometrics Recall that these models are differentfrom the so-called stochastic volatility models like Hestonmodel where the volatility process is driven by an additionalsource of randomness not perfectly correlated with the stockprice innovations

As an application for the particular case of CEV modelwe obtain an approximation of the at-the-money (ATM)implied volatility curve as a function of time and an approx-imation of the implied volatility smile as a function of

the log-moneyness close to the expiry date We use theseapproximations to calibrate the CEV model parameters

2 Preliminaries and Notations

Let 119878 = 119878119905 119905 isin [0 119879] be a positive price process under amarket chosen risk neutral probability that follows the model

119889119878119905 = 119903119878119905119889119905 + 120579 (119878119905) 119878119905119889119882119905 (1)

where 119882 is a standard Brownian motion 119903 ge 0 is theconstant interest rate and 120579 [0infin) rarr [0infin) is afunction of 1198622([0infin)) such that 120579(119878119905) is a square integrablerandom variable that satisfies enough conditions to ensurethe existence and uniqueness of a solution of (1)

The following notation will be used in all the paper(i) We define the Black-Scholes function as a function of119905 isin [0 119879] and 119909 119910 isin [0infin) such that

BS (119905 119909 119910) fl 119909Φ (119889+) minus 119870119890minus119903(119879minus119905)Φ(119889minus) (2)

where Φ(sdot) denotes the cumulative probability func-tion of the standard normal law 119870 and 119879 are strictlypositive constants and

119889plusmn (119910) fl ln (119909119870) + (119903 plusmn 11991022) (119879 minus 119905)119910radic119879 minus 119905 (3)

HindawiInternational Journal of Stochastic AnalysisVolume 2017 Article ID 8019498 16 pageshttpsdoiorg10115520178019498

2 International Journal of Stochastic Analysis

Note that the price of a plain vanilla European callunder the classical Black-Scholes theory is BS(119905 119878119905 120590)where 119878119905 is the price of the underlying process at 119905 120590is the constant volatility119870 is the strike price and 119879 isthe expiry date

(ii) We will denote frequently by 120591 fl 119879 minus 119905 the time tomaturity

(iii) We use in all the paper the notation E119905[sdot] fl E[sdot | F119905]where F119905 119905 ge 0 is the completed natural filtrationof 119878

(iv) In our setting the call option price is given by

119881119905 = 119890minus119903(119879minus119905)E119905 [(119878119879 minus 119870)+] (4)

(v) Recall that from the Feynman-Kac formula the oper-ator

L120579 fl 120597119905 + 12120579 (119878119905)2 1198782119905 1205972119878 + 119903119878119905120597119878 minus 119903 (5)

satisfiesL120579BS(119905 119878119905 120579(119878119905)119878119905) = 0(vi) We define the operators Λ fl 119909120597119909 Γ fl 11990921205972119909 and Γ2 =Γ ∘ Γ In particular we have that

ΓBS (119905 119909 119910) fl 119909119910radic2120587120591 exp(minus1198892+ (119910)2 )

ΛΓBS (119905 119909 119910)fl

119909119910radic2120587120591 exp(minus1198892+ (119910)2 )(1 minus 119889+ (119910)119910radic120591 )

Γ2BS (119905 119909 119910)fl

119909119910radic2120587120591 exp(minus1198892+ (119910)2 ) 1198892+ (119910) minus 119910119889+ (119910)radic120591 minus 1

1199102120591

(6)

Lemma 1 Then for any 119899 ge 2 and for any positive quantities119909 119910 119901 and 119902 one has1003816100381610038161003816119909119901 (ln119909)119902 119909119899120597119899119909119861119878 (119905 119909 119910)1003816100381610038161003816 le 119862

(119910radic120591)119899minus1 (7)

where 119862 is a constant that depends on 119901 119902 and 119899Proof For any 119899 ge 2 we have

119909119899120597119899119909BS (119905 119909 119910) = 119909120601 (119889+)(119910radic120591)119899minus1119875119899minus2 (ln119909 119910radic120591) (8)

where 119875119899minus2 is a polynomial of order 119899minus2 and the exponentialdecreasing on 119909 of the Gaussian kernel compensates thepossible increasing of 119909 and ln 119909

3 A General Decomposition Formula

Here we obtain a general abstract decomposition formula fora certain family of functionals of 119878 that will be the basis of alllater computations

Assume we have a functional of the form

119890minus119903119905119860(119905 119878119905 1205792 (119878119905)) 119861 (119905) (9)

where 119861 is a function of1198622([0 119879]) and119860(119905 119909 119910) is a functionof 119862122([0 119879] times [0infin) times [0infin))

Then we have the following lemma

Lemma 2 (generic decomposition formula) For all 119905 isin[0 119879] one hasE119905 [119890minus119903(119879minus119905)119860(119879 119878119879 1205792 (119878119879)) 119861 (119879)] = 119860 (119905 119878119905 1205792 (119878119905))

sdot 119861 (119905) + E119905 [int119879119905119890minus119903(119906minus119905)119860(119906 119878119906 1205792 (119878119906))

sdot 1198611015840 (119906) 119889119906]+ 119903E119905 [int119879

119905119890minus119903(119906minus119905)1205971205792119860(119906 119878119906 1205792 (119878119906)) 119861 (119906)

sdot (1205971198781205792 (119878119906)) 119878119906119889119906] + 12sdot E119905 [int119879

119905119890minus119903(119906minus119905)1205971205792119860(119906 119878119906 1205792 (119878119906)) 119861 (119906)

sdot (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198782119906119889119906] + 12sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721205792119860(119906 119878119906 1205792 (119878119906)) 119861 (119906)

sdot (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198782119906119889119906]+ E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792119860(119906 119878119906 1205792 (119878119906)) 119861 (119906)

sdot (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198782119906119889119906]

(10)

Proof Applying the Ito formula to process 119890minus119903119905119860(119905 1198781199051205792(119878119905))119861(119905) we obtain119890minus119903119879119860(119879 119878119879 1205792 (119878119879)) 119861 (119879) = 119890minus119903119905119860(119905 119878119905 1205792 (119878119905))

sdot 119861 (119905) minus 119903int119879119905119890minus119903119906119860(119906 119878119906 1205792 (119878119906)) 119861 (119906) 119889119906

+ int119879119905119890minus119903119906120597119906119860(119906 119878119906 1205792 (119878119906)) 119861 (119906) 119889119906

+ int119879119905119890minus119903119906120597119878119860(119906 119878119906 1205792 (119878119906)) 119861 (119906) 119889119878119906

+ int119879119905119890minus1199031199061205971205792119860(119906 119878119906 1205792 (119878119906))

International Journal of Stochastic Analysis 3

sdot 119861 (119906) 1198891205792 (119878119906) + int119879119905119890minus119903119906119860(119906 119878119906 1205792 (119878119906))

sdot 1198611015840 (119906) 119889119906 + 12 int119879119905119890minus1199031199061205972119878119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [119878 119878]119906 + 12sdot int119879119905119890minus11990311990612059721205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [1205792 (119878) 1205792 (119878)]119906

+ int119879119905119890minus11990311990612059721198781205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [119878 1205792 (119878)]119906

(11)

Now applying Feynman-Kac formula for 1205792(119878119905)1198782119905 multi-plying by 119890119903119905 and taking conditional expectations we obtain

E119905 [119890minus119903(119879minus119905)119860(119879 119878119879 1205792 (119878119879)) 119861 (119879)] = 119860 (119905 119878119905 1205792 (119878119905))sdot 119861 (119905) + E119905 [int119879

119905119890minus119903(119906minus119905)119860(119906 119878119906 1205792 (119878119906))

sdot 1198611015840 (119906) 119889119906]+ E119905 [int119879

119905119890minus119903(119906minus119905)1205971205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 1198891205792 (119878119906)] + 12sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [1205792 (119878) 1205792 (119878)]119906]

+ E119905 [int119879119905119890minus119903(119906minus119905)12059721198781205792119860(119906 119878119906 1205792 (119878119906)) 119861 (119906) 120579 (119878119906)

sdot 119878119906119889 [119882 1205792 (119878)]119906]

(12)

On the other hand using Ito calculus rules it is easy tosee that

1198891205792 (119878119905) = 1205971198781205792 (119878119905) 119903119878119905119889119905 + 1205971198781205792 (119878119905) 120579 (119878119905) 119878119905119889119882119905+ 1212059721198781205792 (119878119905) 1205792 (119878119905) 1198782119905119889119905

(13)

Finally substituting this expression in (12) we finish theproof

For the Black-Scholes function previous lemma reducesto the following corollary

Corollary 3 (BS decomposition formula) For all 119905 isin [0 119879]one has

E119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120579 (119878119879))] = 119861119878 (119905 119878119905 120579 (119878119905)) + 1199032sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)

sdot (1205971198781205792 (119878119906)) 119878119906119889119906] + 14sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)

sdot (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)2

sdot (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198782119906119889119906] + 12sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)

sdot (1205971198781205792 (119878119906)) 1205792 (119878119906) 119878119906119889119906]

(14)

Proof Applying Lemma 2 to119860(119905 119878119905 1205792(119878119905)) fl BS(119905 119878119905 120579(119878119905))and 119861(119905) equiv 1 and using equalities

1205971205792BS (119905 119878119905 120579 (119878119905)) = 119879 minus 1199052 1198782119905 1205972119878BS (119905 119878119905 120579 (119878119905)) 12059721205792BS (119905 119878119905 120579 (119878119905))

= (119879 minus 119905)24 1198782119905 1205972119878 (1198782119905 1205972119878BS (119905 119878119905 120579 (119878119905)))(15)

the corollary follows straightforward Note that to apply Itoformula to Black-Scholes function because the derivatives ofthis function are not bounded we have to use an approxima-tion to the identity and the dominated convergence theoremas it is done for example in [3] For simplicity we skip thismollifying argument across the paper

Remark 4 For clarity in the following we will refer to termsof the previous decomposition as

E119905 [119890minus119903(119879minus119905)BS (119879 119878119879 120579 (119878119879))]= BS (119905 119878119905 120579 (119878119905)) + (I) + (II) + (III) + (IV) (16)

Remark 5 In [4] an alternative formula that can be usedfor local volatility models is proved The formula presentedin [4] uses as a base function function BS(119905 119878119905 120590) but thisformula is numerically worse than the new formula presentedhere that uses as a base function BS(119905 119878119905 120579(119878119905)) This happensbecause in the formula presented in [4] the volatility is putinto the approximated term instead of keeping it on the

4 International Journal of Stochastic Analysis

Black-Scholes term as we do here It is precisely becausevolatility is a deterministic function of the underlying assetprice that we can do that

4 Approximation Formula

In this section we obtain an approximation formula to plainvanilla call price by approximating terms (I)ndash(IV) The mainidea is to use again Lemma 2 to estimate the errors

Theorem6 (BS decomposition formula with error term) Forall 119905 isin [0 119879] one hasE119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120579 (119878119879))] = 119861119878 (119905 119878119905 120579 (119878119905)) + 14

sdot 119903 (1205971198781205792 (119878119905)) 119878119905Γ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2+ 18 (12059721198781205792 (119878119905)) 1205792 (119878119905) 1198782119905 Γ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2+ 124 (1205971198781205792 (119878119905))2 1205792 (119878119905) 1198782119905 Γ2119861119878 (119905 119878119905 120579 (119878119905))sdot (119879 minus 119905)3 + 14 (1205971198781205792 (119878119905)) 1205792 (119878119905)sdot 119878119905ΛΓ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 + Ω

(17)

whereΩ is an error Terms of Ω are written in Appendix A

Proof We apply Lemma 2 to terms (I)ndash(IV) Concretelyfunctions 119860 and 119861 in every case are

(I)

119860(119905 119878119905 1205792 (119878119905)) = (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) 119861119905 = 1199032 int119879

119905(119879 minus 119906) 119889119906 (18)

(II)

119860(119905 119878119905 1205792 (119878119905))= (12059721198781205792 (119878119905)) 1205792 (119878119905) 1198782119905 ΓBS (119905 119878119905 120579 (119878119905))

119861119905 = 14 int119879119905(119879 minus 119906) 119889119906

(19)

(III)

119860(119905 119878119905 1205792 (119878119905))= (1205971198781205792 (119878119905))2 1205792 (119878119905) 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905))

119861119905 = 18 int119879119905(119879 minus 119906)2 119889119906

(20)

(IV)

119860(119905 119878119905 1205792 (119878119905))= (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905 119878119905 120579 (119878119905))

119861119905 = 12 int119879119905(119879 minus 119906) 119889119906

(21)

5 CEV Model

The constant elasticity of variance (CEV) model is a diffusionprocess that solves the stochastic differential equation

119889119878119905 = 119903119878119905119889119905 + 120590119878120573119905 119889119882119905 (22)

Note that writing 120579(119878119905) fl 120590119878120573minus1119905 CEV model can be seenas a local volatility model This model introduced in [5] isone of the first alternatives to Black-Scholes point of view thatappeared in the literature The parameter 120573 ge 0 is called theelasticity of the volatility and 120590 ge 0 is a scale parameter Notethat for 120573 = 1 the model reduces to Osborne-Samuelsonmodel for 120573 = 0 the model reduces to Bachelier model andfor 120573 = 12 the model reduces to Cox-Ingersoll-Ross modelParameter 120573 controls the steepness of the skew exhibited bythe implied volatility

There exists a closed form formula for call options see[5 6] An approximated formula is given in [7]

51 Approximation of the CEV Model Applying Corollary 3to CEV model we obtain the following

Corollary 7 (CEV exact formula) For all 119905 isin [0 119879] one hasE119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120590119878120573minus1119879 )] = 119861119878 (119905 119878119905 120590119878120573minus1119905 )

+ 119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120590119878120573minus1119906 )

sdot (119879 minus 119906) 12059021198782(120573minus1)119905 119889119906] + (120573 minus 1) (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)

sdot 12059041198784(120573minus1)119906 119889119906] + (120573 minus 1)22

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)2

sdot 12059061198786(120573minus1)119906 119889119906] + (120573 minus 1)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)

sdot 12059041198784(120573minus1)119906 119889119906]

(23)

We will write

E119905 [119890minus119903(119879minus119905)BS (119879 119878119879 120590119878120573minus1119879 )]= BS (119905 119878119905 120590119878120573minus1119905 ) + (ICEV) + (IICEV) + (IIICEV)

+ (IVCEV) (24)

The exact formula can be difficult to use in practice so wewill use the following approximation

International Journal of Stochastic Analysis 5

Table 1 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 02882882 02882884 minus192119864 minus 07 02882019 864119864 minus 051 10103060 10103070 minus978119864 minus 07 10100377 268119864 minus 0425 24709883 24709894 minus104119864 minus 06 24708310 157119864 minus 045 48771276 48771278 minus222119864 minus 07 48771099 177119864 minus 05

Table 2 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 05356736 05356765 minus289119864 minus 06 05354323 241119864 minus 041 13886303 13886529 minus226119864 minus 05 13868801 175119864 minus 0325 28506826 28507669 minus842119864 minus 05 28450032 568119864 minus 035 51658348 51660433 minus209119864 minus 04 51543092 115119864 minus 02

Table 3 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 13887209 13887438 minus230119864 minus 05 13883284 392119864 minus 041 30389972 30391797 minus183119864 minus 04 30359001 310119864 minus 0325 52954739 52961870 minus713119864 minus 04 52835621 119119864 minus 025 82781049 82800813 minus198119864 minus 03 82459195 322119864 minus 02

Corollary 8 (CEV approximation formula) For all 119905 isin [0 119879]one has

E119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120590119878120573minus1119879 )] = 119861119878 (119905 119878119905 120590119878120573minus1119905 )+ 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2+ 14 (120573 minus 1) (2120573 minus 3) 12059041198784(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)3 + 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + Ω

(25)

whereΩ is an error Terms ofΩ are written in Appendix B Wehave that Ω le (120573 minus 1)2Π(119905 119879 119903 120590 120573) and Π is an increasingfunction on every parameter

Proof Theproof is a direct consequence of applying Lemma 1to (ICEV)ndash(IVCEV) In Appendix C the upper-bounds forevery term are given

52 Numerical Analysis of the Approximation for the CEVCase In this section we compare our numerically approx-imated price of a CEV call option with the following differentpricing methods

(i) The exact formula see [5 6 8] The Matlab code isavailable in [9]

(ii) The Singular Perturbation Technique see [7]

The results for a call option with parameters 120573 = 0251198780 = 100 119870 = 100 120590 = 20 and 119903 = 1 are presented inTable 1

The results in the case that 120573 = 050 are presented inTable 2

The results in the case that 120573 = 075 are presented inTable 3

Finally the results in the case that 120573 = 090 are presentedin Table 4

Note that the new approximation is more accurate thanthe approximation obtained in [7]

In Figure 1 we plot the surface of errors between the exactformula and our approximation

We calculate also the speed time of execution (in seconds)of every method running the function timeit of Matlab 1000times The computer used is an Intel Core i7 CPU Q740

6 International Journal of Stochastic Analysis

Table 4 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 26404164 26404455 minus292119864 minus 05 26401025 314119864 minus 041 55191736 55194053 minus232119864 minus 04 55166821 249119864 minus 0325 91446125 91455159 minus903119864 minus 04 91349142 970119864 minus 035 135553379 135578351 minus250119864 minus 03 135286009 267119864 minus 02

Table 5 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 minus 119905 = 5 and 119903 = 1

Measure Exact formula Approximation HWAverage 256119864 minus 02 173119864 minus 04 167119864 minus 04Standard deviation 303119864 minus 03 286119864 minus 05 252119864 minus 05Max 468119864 minus 02 365119864 minus 04 367119864 minus 04Min 242119864 minus 02 164119864 minus 04 159119864 minus 04

173GHz 173GHz with 4GB of RAM with a Windows 10(times64) The results are presented in Table 5

We observe that singular perturbation method is thefastest method to calculate the price of CEV call option Themethod developed in this work is a little more expensive incomputation time But to compute the exact price is muchmore expensive than any of the other twomethods Note thatin our method we also are able to calculate at the same timethe price and the Gamma of the log-normal price

6 The Approximated Implied VolatilitySurface under CEV Model

In the above section we have computed a bound for theerror between the exact and the approximated pricing for-mulas for the CEV model Now we are going to derive anapproximated implied volatility surface of second order in thelog-moneyness This approximated implied volatility surfacecan help us to understand better the volatility dynamicsMoreover we obtain an approximation of the ATM impliedvolatility dynamics

61 Deriving an Approximated Implied Volatility Surface forthe CEV Model In this section for simplicity and withoutlosing generality we assume 119905 = 0 So 119879 = 120591 denotes timeto maturity The price of an European call option with strike119870 and maturity 119879 is an observable quantity which will bereferred to as 119875obs

0 = 119875obs(119870 119879) The implied volatility isdefined as the value 119868(119879119870) that makes

BS (0 1198780 119868 (119879119870)) = 119875obs0 (26)

Using the results from the previous section we are goingto derive an approximation to the implied volatility as in [10]We use the idea to expand the function with respect to anasymptotic sequence 120575119896infin119896=0 converging to 0 Thus we canwrite

119891 = 1198910 + 1205751198911 + 12057521198912 + 119874 (1205753) (27)

and assuming 120573 isin (0 2) we can choose 120575 = 120573 minus 1 Then wecan expand 119868(119879119870) with respect to this scale as

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870)+ 119874 ((120573 minus 1)3) (28)

and write

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) (29)

Let V0 fl 120590119878120573minus10 Write BS(V0) as a shorthand for BS(01198780 V0)We can rewrite Corollary 8 as

(0 1198780 V0) = BS (V0) + 14 (120573 minus 1)sdot 119879V0 (2119903 + V20 [1 minus 2119889+

V0radic119879])sdot 120597120590BS (V0) + 16 (120573 minus 1)2sdot V30119879 ([1198892+ minus V0radic119879119889+ + 2])sdot 120597120590BS (V0)

(30)

On the other hand we can consider the Taylor expansionof BS(0 1198780 119868(119879119870)) around V0We have that

0 = BS (V0) + 120597120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot) + 12sdot 1205972120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot)2+ sdot sdot sdot

(31)

International Journal of Stochastic Analysis 7

01020304050607080917080K

90100

110120

1

minus1

minus05

15

05

0

130

times10minus4

Figure 1 Error surface between exact formula and our approxima-tion for 1198780 = 100 120590 = 20 and 119903 = 5

and this expression can be rewritten as

BS (119868 (119879119870)) = BS (V0) + (120573 minus 1) 120597120590BS (V0) 1198681 (119879119870)+ (120573 minus 1)2 120597120590BS (V0) 1198682 (119879119870)+ 119874 ((120573 minus 1)2)

(32)

Then equating this expression to 0 we have1198681 (119879119870) = 119879V04 (2119903 + V20 [1 minus 2119889+

V0radic119879])

1198682 (119879119870) = 119879V306 (1198892+ minus V0radic119879119889+ + 2) (33)

Note that 1198681(119879119870) is linear with respect to the log-moneyness while 1198682(119879119870) is quadraticRemark 9 Note that the pricing formula has an error of119874((120573 minus 1)2) as we have proved in Corollary 8 and this istranslated into an error of119874((120573minus1)2) into our approximationof the implied volatility The quadratic term of the volatilityshape is not accurate

We calculate now the short time behavior of the approx-imated implied volatility 119868(119879119870) We write the approximatedequations in terms of 1minus120573 because the case 120573 lt 1 is the mostinteresting and in terms of the log-moneyness ln119870 minus ln 1198780Lemma 10 For 119879 close to 0 one has

119868 (119879119870) asymp V0 minus V02 (1 minus 120573) (ln119870 minus ln 1198780)+ V06 (1 minus 120573)2 (ln119870 minus ln 1198780)2

(34)

Proof Note that

lim119879rarr0

1198681 (119879119870) = V02 (ln119870 minus ln 1198780) lim119879rarr0

1198682 (119879119870) = lim119879rarr0

V301198796 (1198892+ minus V0radic119879119889+ + 2)= V06 (ln119870 minus ln 1198780)2

(35)

Remark 11 Note that (34) is a parabolic equation in the log-moneyness Also from the above expression it is easy to seethat the slope with respect to ln119870 is negative when 119870 lt1198780 exp(32(1 minus120573)) and positive when119870 gt 1198780 exp(32(1 minus120573))showing that the implied volatility for short times tomaturityis smile-shaped This is consistent with the result in [11]Furthermore there is a minimum of the implied volatilitywith respect to ln119870 attained at 119870 = 1198780 exp(32(1 minus 120573))Remark 12 Note that in stochastic volatility models theimplied volatility depends homogeneously on the pair (119878 119870)and in fact it is a function of the log-moneyness ln(1198780119870)As extensively discussed in [12] and exemplified for GARCHoption pricing in [13] this homogeneity property is at oddswith any type of GARCH option pricing We found also thisphenomenon in the quadratic expansion (34)

The behavior of the approximated implied volatility whenthe option is ATM is easy to obtain

119868 (119879119870) = V0 + (V0119903 (120573 minus 1)2 + V30 (120573 minus 1)2

3 )119879

minus (120573 minus 1)2 V6024 1198792(36)

62 Numerical Analysis of the Approximation of the ImpliedVolatility for the CEV Case In this section we com-pare numerically our approximated implied volatilities withimplied volatility computed from call option prices calculatedwith the exact formula and with the ones obtained using thefollowing formula obtained in [7]

119868 (119879119870) = 1205901198911minus120573av

[1 + (1 minus 120573) (2 + 120573)24 (1198650 minus 119870119891av )2

+ (1 minus 120573)224 1205902119879

1198912minus2120573av]

(37)

where 119891av = (12)(1198650 minus 119870) and 1198650 is the forward priceIn Figure 2 we can see that the implied volatility dynam-

ics behaves well for long dated maturities and short datedmaturities when 120573 is close to 1 When this is not the case theformula behaves well at-the-money but the error increases farfrom the ATM value This behavior is a consequence of thequadratic error of our approximation

Comparing the ATM volatility structure we have thefollowing graphics

In Figure 3 we observe that for ATM options theapproximated implied volatility surface fits really well the realimplied volatility structure

Now we put the implied volatility approximation foundin (34) into Black-Scholes formula and compare the obtainedresults with Hagan and Woodward results The results for acall option with parameters 120573 = 025 1198780 = 100 119870 = 100120590 = 20 and 119903 = 1 are presented in Table 6

The results in the case that 120573 = 050 are presented inTable 7

8 International Journal of Stochastic Analysis

0019

00195

002

00205

0021

Impl

ied

vola

tility

10610210098 108 11090 10492 9694K

= 05 T = 1

80 90 100 110 120 13070K

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 1

Exact formulaApproximationHW

92 94 96 98 100 102 104 106 108 11090K

0018

00185

0019

00195

002

00205

0021

00215

Impl

ied

vola

tility

= 05 T = 5

Exact formulaApproximationHW

80 90 100 110 120 13070K

0123

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 5

Figure 2 Comparative of implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

001750018

001850019

00195002

002050021

00215

Impl

ied

vola

tility

05 1 15 4525 3 35 50 2 4T

ATM = 05

Exact formulaApproximation

HWApproximation ATM

012301235

012401245

012501255

012601265

Impl

ied

vola

tility

05 2 41 25 453515 50 3T

ATM = 09

Figure 3 Comparative of ATM implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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International Journal of Stochastic Analysis 3

sdot 119861 (119906) 1198891205792 (119878119906) + int119879119905119890minus119903119906119860(119906 119878119906 1205792 (119878119906))

sdot 1198611015840 (119906) 119889119906 + 12 int119879119905119890minus1199031199061205972119878119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [119878 119878]119906 + 12sdot int119879119905119890minus11990311990612059721205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [1205792 (119878) 1205792 (119878)]119906

+ int119879119905119890minus11990311990612059721198781205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [119878 1205792 (119878)]119906

(11)

Now applying Feynman-Kac formula for 1205792(119878119905)1198782119905 multi-plying by 119890119903119905 and taking conditional expectations we obtain

E119905 [119890minus119903(119879minus119905)119860(119879 119878119879 1205792 (119878119879)) 119861 (119879)] = 119860 (119905 119878119905 1205792 (119878119905))sdot 119861 (119905) + E119905 [int119879

119905119890minus119903(119906minus119905)119860(119906 119878119906 1205792 (119878119906))

sdot 1198611015840 (119906) 119889119906]+ E119905 [int119879

119905119890minus119903(119906minus119905)1205971205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 1198891205792 (119878119906)] + 12sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721205792119860(119906 119878119906 1205792 (119878119906))

sdot 119861 (119906) 119889 [1205792 (119878) 1205792 (119878)]119906]

+ E119905 [int119879119905119890minus119903(119906minus119905)12059721198781205792119860(119906 119878119906 1205792 (119878119906)) 119861 (119906) 120579 (119878119906)

sdot 119878119906119889 [119882 1205792 (119878)]119906]

(12)

On the other hand using Ito calculus rules it is easy tosee that

1198891205792 (119878119905) = 1205971198781205792 (119878119905) 119903119878119905119889119905 + 1205971198781205792 (119878119905) 120579 (119878119905) 119878119905119889119882119905+ 1212059721198781205792 (119878119905) 1205792 (119878119905) 1198782119905119889119905

(13)

Finally substituting this expression in (12) we finish theproof

For the Black-Scholes function previous lemma reducesto the following corollary

Corollary 3 (BS decomposition formula) For all 119905 isin [0 119879]one has

E119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120579 (119878119879))] = 119861119878 (119905 119878119905 120579 (119878119905)) + 1199032sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)

sdot (1205971198781205792 (119878119906)) 119878119906119889119906] + 14sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)

sdot (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)2

sdot (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198782119906119889119906] + 12sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ119861119878 (119906 119878119906 120579 (119878119906)) (119879 minus 119906)

sdot (1205971198781205792 (119878119906)) 1205792 (119878119906) 119878119906119889119906]

(14)

Proof Applying Lemma 2 to119860(119905 119878119905 1205792(119878119905)) fl BS(119905 119878119905 120579(119878119905))and 119861(119905) equiv 1 and using equalities

1205971205792BS (119905 119878119905 120579 (119878119905)) = 119879 minus 1199052 1198782119905 1205972119878BS (119905 119878119905 120579 (119878119905)) 12059721205792BS (119905 119878119905 120579 (119878119905))

= (119879 minus 119905)24 1198782119905 1205972119878 (1198782119905 1205972119878BS (119905 119878119905 120579 (119878119905)))(15)

the corollary follows straightforward Note that to apply Itoformula to Black-Scholes function because the derivatives ofthis function are not bounded we have to use an approxima-tion to the identity and the dominated convergence theoremas it is done for example in [3] For simplicity we skip thismollifying argument across the paper

Remark 4 For clarity in the following we will refer to termsof the previous decomposition as

E119905 [119890minus119903(119879minus119905)BS (119879 119878119879 120579 (119878119879))]= BS (119905 119878119905 120579 (119878119905)) + (I) + (II) + (III) + (IV) (16)

Remark 5 In [4] an alternative formula that can be usedfor local volatility models is proved The formula presentedin [4] uses as a base function function BS(119905 119878119905 120590) but thisformula is numerically worse than the new formula presentedhere that uses as a base function BS(119905 119878119905 120579(119878119905)) This happensbecause in the formula presented in [4] the volatility is putinto the approximated term instead of keeping it on the

4 International Journal of Stochastic Analysis

Black-Scholes term as we do here It is precisely becausevolatility is a deterministic function of the underlying assetprice that we can do that

4 Approximation Formula

In this section we obtain an approximation formula to plainvanilla call price by approximating terms (I)ndash(IV) The mainidea is to use again Lemma 2 to estimate the errors

Theorem6 (BS decomposition formula with error term) Forall 119905 isin [0 119879] one hasE119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120579 (119878119879))] = 119861119878 (119905 119878119905 120579 (119878119905)) + 14

sdot 119903 (1205971198781205792 (119878119905)) 119878119905Γ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2+ 18 (12059721198781205792 (119878119905)) 1205792 (119878119905) 1198782119905 Γ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2+ 124 (1205971198781205792 (119878119905))2 1205792 (119878119905) 1198782119905 Γ2119861119878 (119905 119878119905 120579 (119878119905))sdot (119879 minus 119905)3 + 14 (1205971198781205792 (119878119905)) 1205792 (119878119905)sdot 119878119905ΛΓ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 + Ω

(17)

whereΩ is an error Terms of Ω are written in Appendix A

Proof We apply Lemma 2 to terms (I)ndash(IV) Concretelyfunctions 119860 and 119861 in every case are

(I)

119860(119905 119878119905 1205792 (119878119905)) = (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) 119861119905 = 1199032 int119879

119905(119879 minus 119906) 119889119906 (18)

(II)

119860(119905 119878119905 1205792 (119878119905))= (12059721198781205792 (119878119905)) 1205792 (119878119905) 1198782119905 ΓBS (119905 119878119905 120579 (119878119905))

119861119905 = 14 int119879119905(119879 minus 119906) 119889119906

(19)

(III)

119860(119905 119878119905 1205792 (119878119905))= (1205971198781205792 (119878119905))2 1205792 (119878119905) 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905))

119861119905 = 18 int119879119905(119879 minus 119906)2 119889119906

(20)

(IV)

119860(119905 119878119905 1205792 (119878119905))= (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905 119878119905 120579 (119878119905))

119861119905 = 12 int119879119905(119879 minus 119906) 119889119906

(21)

5 CEV Model

The constant elasticity of variance (CEV) model is a diffusionprocess that solves the stochastic differential equation

119889119878119905 = 119903119878119905119889119905 + 120590119878120573119905 119889119882119905 (22)

Note that writing 120579(119878119905) fl 120590119878120573minus1119905 CEV model can be seenas a local volatility model This model introduced in [5] isone of the first alternatives to Black-Scholes point of view thatappeared in the literature The parameter 120573 ge 0 is called theelasticity of the volatility and 120590 ge 0 is a scale parameter Notethat for 120573 = 1 the model reduces to Osborne-Samuelsonmodel for 120573 = 0 the model reduces to Bachelier model andfor 120573 = 12 the model reduces to Cox-Ingersoll-Ross modelParameter 120573 controls the steepness of the skew exhibited bythe implied volatility

There exists a closed form formula for call options see[5 6] An approximated formula is given in [7]

51 Approximation of the CEV Model Applying Corollary 3to CEV model we obtain the following

Corollary 7 (CEV exact formula) For all 119905 isin [0 119879] one hasE119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120590119878120573minus1119879 )] = 119861119878 (119905 119878119905 120590119878120573minus1119905 )

+ 119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120590119878120573minus1119906 )

sdot (119879 minus 119906) 12059021198782(120573minus1)119905 119889119906] + (120573 minus 1) (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)

sdot 12059041198784(120573minus1)119906 119889119906] + (120573 minus 1)22

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)2

sdot 12059061198786(120573minus1)119906 119889119906] + (120573 minus 1)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)

sdot 12059041198784(120573minus1)119906 119889119906]

(23)

We will write

E119905 [119890minus119903(119879minus119905)BS (119879 119878119879 120590119878120573minus1119879 )]= BS (119905 119878119905 120590119878120573minus1119905 ) + (ICEV) + (IICEV) + (IIICEV)

+ (IVCEV) (24)

The exact formula can be difficult to use in practice so wewill use the following approximation

International Journal of Stochastic Analysis 5

Table 1 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 02882882 02882884 minus192119864 minus 07 02882019 864119864 minus 051 10103060 10103070 minus978119864 minus 07 10100377 268119864 minus 0425 24709883 24709894 minus104119864 minus 06 24708310 157119864 minus 045 48771276 48771278 minus222119864 minus 07 48771099 177119864 minus 05

Table 2 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 05356736 05356765 minus289119864 minus 06 05354323 241119864 minus 041 13886303 13886529 minus226119864 minus 05 13868801 175119864 minus 0325 28506826 28507669 minus842119864 minus 05 28450032 568119864 minus 035 51658348 51660433 minus209119864 minus 04 51543092 115119864 minus 02

Table 3 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 13887209 13887438 minus230119864 minus 05 13883284 392119864 minus 041 30389972 30391797 minus183119864 minus 04 30359001 310119864 minus 0325 52954739 52961870 minus713119864 minus 04 52835621 119119864 minus 025 82781049 82800813 minus198119864 minus 03 82459195 322119864 minus 02

Corollary 8 (CEV approximation formula) For all 119905 isin [0 119879]one has

E119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120590119878120573minus1119879 )] = 119861119878 (119905 119878119905 120590119878120573minus1119905 )+ 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2+ 14 (120573 minus 1) (2120573 minus 3) 12059041198784(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)3 + 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + Ω

(25)

whereΩ is an error Terms ofΩ are written in Appendix B Wehave that Ω le (120573 minus 1)2Π(119905 119879 119903 120590 120573) and Π is an increasingfunction on every parameter

Proof Theproof is a direct consequence of applying Lemma 1to (ICEV)ndash(IVCEV) In Appendix C the upper-bounds forevery term are given

52 Numerical Analysis of the Approximation for the CEVCase In this section we compare our numerically approx-imated price of a CEV call option with the following differentpricing methods

(i) The exact formula see [5 6 8] The Matlab code isavailable in [9]

(ii) The Singular Perturbation Technique see [7]

The results for a call option with parameters 120573 = 0251198780 = 100 119870 = 100 120590 = 20 and 119903 = 1 are presented inTable 1

The results in the case that 120573 = 050 are presented inTable 2

The results in the case that 120573 = 075 are presented inTable 3

Finally the results in the case that 120573 = 090 are presentedin Table 4

Note that the new approximation is more accurate thanthe approximation obtained in [7]

In Figure 1 we plot the surface of errors between the exactformula and our approximation

We calculate also the speed time of execution (in seconds)of every method running the function timeit of Matlab 1000times The computer used is an Intel Core i7 CPU Q740

6 International Journal of Stochastic Analysis

Table 4 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 26404164 26404455 minus292119864 minus 05 26401025 314119864 minus 041 55191736 55194053 minus232119864 minus 04 55166821 249119864 minus 0325 91446125 91455159 minus903119864 minus 04 91349142 970119864 minus 035 135553379 135578351 minus250119864 minus 03 135286009 267119864 minus 02

Table 5 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 minus 119905 = 5 and 119903 = 1

Measure Exact formula Approximation HWAverage 256119864 minus 02 173119864 minus 04 167119864 minus 04Standard deviation 303119864 minus 03 286119864 minus 05 252119864 minus 05Max 468119864 minus 02 365119864 minus 04 367119864 minus 04Min 242119864 minus 02 164119864 minus 04 159119864 minus 04

173GHz 173GHz with 4GB of RAM with a Windows 10(times64) The results are presented in Table 5

We observe that singular perturbation method is thefastest method to calculate the price of CEV call option Themethod developed in this work is a little more expensive incomputation time But to compute the exact price is muchmore expensive than any of the other twomethods Note thatin our method we also are able to calculate at the same timethe price and the Gamma of the log-normal price

6 The Approximated Implied VolatilitySurface under CEV Model

In the above section we have computed a bound for theerror between the exact and the approximated pricing for-mulas for the CEV model Now we are going to derive anapproximated implied volatility surface of second order in thelog-moneyness This approximated implied volatility surfacecan help us to understand better the volatility dynamicsMoreover we obtain an approximation of the ATM impliedvolatility dynamics

61 Deriving an Approximated Implied Volatility Surface forthe CEV Model In this section for simplicity and withoutlosing generality we assume 119905 = 0 So 119879 = 120591 denotes timeto maturity The price of an European call option with strike119870 and maturity 119879 is an observable quantity which will bereferred to as 119875obs

0 = 119875obs(119870 119879) The implied volatility isdefined as the value 119868(119879119870) that makes

BS (0 1198780 119868 (119879119870)) = 119875obs0 (26)

Using the results from the previous section we are goingto derive an approximation to the implied volatility as in [10]We use the idea to expand the function with respect to anasymptotic sequence 120575119896infin119896=0 converging to 0 Thus we canwrite

119891 = 1198910 + 1205751198911 + 12057521198912 + 119874 (1205753) (27)

and assuming 120573 isin (0 2) we can choose 120575 = 120573 minus 1 Then wecan expand 119868(119879119870) with respect to this scale as

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870)+ 119874 ((120573 minus 1)3) (28)

and write

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) (29)

Let V0 fl 120590119878120573minus10 Write BS(V0) as a shorthand for BS(01198780 V0)We can rewrite Corollary 8 as

(0 1198780 V0) = BS (V0) + 14 (120573 minus 1)sdot 119879V0 (2119903 + V20 [1 minus 2119889+

V0radic119879])sdot 120597120590BS (V0) + 16 (120573 minus 1)2sdot V30119879 ([1198892+ minus V0radic119879119889+ + 2])sdot 120597120590BS (V0)

(30)

On the other hand we can consider the Taylor expansionof BS(0 1198780 119868(119879119870)) around V0We have that

0 = BS (V0) + 120597120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot) + 12sdot 1205972120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot)2+ sdot sdot sdot

(31)

International Journal of Stochastic Analysis 7

01020304050607080917080K

90100

110120

1

minus1

minus05

15

05

0

130

times10minus4

Figure 1 Error surface between exact formula and our approxima-tion for 1198780 = 100 120590 = 20 and 119903 = 5

and this expression can be rewritten as

BS (119868 (119879119870)) = BS (V0) + (120573 minus 1) 120597120590BS (V0) 1198681 (119879119870)+ (120573 minus 1)2 120597120590BS (V0) 1198682 (119879119870)+ 119874 ((120573 minus 1)2)

(32)

Then equating this expression to 0 we have1198681 (119879119870) = 119879V04 (2119903 + V20 [1 minus 2119889+

V0radic119879])

1198682 (119879119870) = 119879V306 (1198892+ minus V0radic119879119889+ + 2) (33)

Note that 1198681(119879119870) is linear with respect to the log-moneyness while 1198682(119879119870) is quadraticRemark 9 Note that the pricing formula has an error of119874((120573 minus 1)2) as we have proved in Corollary 8 and this istranslated into an error of119874((120573minus1)2) into our approximationof the implied volatility The quadratic term of the volatilityshape is not accurate

We calculate now the short time behavior of the approx-imated implied volatility 119868(119879119870) We write the approximatedequations in terms of 1minus120573 because the case 120573 lt 1 is the mostinteresting and in terms of the log-moneyness ln119870 minus ln 1198780Lemma 10 For 119879 close to 0 one has

119868 (119879119870) asymp V0 minus V02 (1 minus 120573) (ln119870 minus ln 1198780)+ V06 (1 minus 120573)2 (ln119870 minus ln 1198780)2

(34)

Proof Note that

lim119879rarr0

1198681 (119879119870) = V02 (ln119870 minus ln 1198780) lim119879rarr0

1198682 (119879119870) = lim119879rarr0

V301198796 (1198892+ minus V0radic119879119889+ + 2)= V06 (ln119870 minus ln 1198780)2

(35)

Remark 11 Note that (34) is a parabolic equation in the log-moneyness Also from the above expression it is easy to seethat the slope with respect to ln119870 is negative when 119870 lt1198780 exp(32(1 minus120573)) and positive when119870 gt 1198780 exp(32(1 minus120573))showing that the implied volatility for short times tomaturityis smile-shaped This is consistent with the result in [11]Furthermore there is a minimum of the implied volatilitywith respect to ln119870 attained at 119870 = 1198780 exp(32(1 minus 120573))Remark 12 Note that in stochastic volatility models theimplied volatility depends homogeneously on the pair (119878 119870)and in fact it is a function of the log-moneyness ln(1198780119870)As extensively discussed in [12] and exemplified for GARCHoption pricing in [13] this homogeneity property is at oddswith any type of GARCH option pricing We found also thisphenomenon in the quadratic expansion (34)

The behavior of the approximated implied volatility whenthe option is ATM is easy to obtain

119868 (119879119870) = V0 + (V0119903 (120573 minus 1)2 + V30 (120573 minus 1)2

3 )119879

minus (120573 minus 1)2 V6024 1198792(36)

62 Numerical Analysis of the Approximation of the ImpliedVolatility for the CEV Case In this section we com-pare numerically our approximated implied volatilities withimplied volatility computed from call option prices calculatedwith the exact formula and with the ones obtained using thefollowing formula obtained in [7]

119868 (119879119870) = 1205901198911minus120573av

[1 + (1 minus 120573) (2 + 120573)24 (1198650 minus 119870119891av )2

+ (1 minus 120573)224 1205902119879

1198912minus2120573av]

(37)

where 119891av = (12)(1198650 minus 119870) and 1198650 is the forward priceIn Figure 2 we can see that the implied volatility dynam-

ics behaves well for long dated maturities and short datedmaturities when 120573 is close to 1 When this is not the case theformula behaves well at-the-money but the error increases farfrom the ATM value This behavior is a consequence of thequadratic error of our approximation

Comparing the ATM volatility structure we have thefollowing graphics

In Figure 3 we observe that for ATM options theapproximated implied volatility surface fits really well the realimplied volatility structure

Now we put the implied volatility approximation foundin (34) into Black-Scholes formula and compare the obtainedresults with Hagan and Woodward results The results for acall option with parameters 120573 = 025 1198780 = 100 119870 = 100120590 = 20 and 119903 = 1 are presented in Table 6

The results in the case that 120573 = 050 are presented inTable 7

8 International Journal of Stochastic Analysis

0019

00195

002

00205

0021

Impl

ied

vola

tility

10610210098 108 11090 10492 9694K

= 05 T = 1

80 90 100 110 120 13070K

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 1

Exact formulaApproximationHW

92 94 96 98 100 102 104 106 108 11090K

0018

00185

0019

00195

002

00205

0021

00215

Impl

ied

vola

tility

= 05 T = 5

Exact formulaApproximationHW

80 90 100 110 120 13070K

0123

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 5

Figure 2 Comparative of implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

001750018

001850019

00195002

002050021

00215

Impl

ied

vola

tility

05 1 15 4525 3 35 50 2 4T

ATM = 05

Exact formulaApproximation

HWApproximation ATM

012301235

012401245

012501255

012601265

Impl

ied

vola

tility

05 2 41 25 453515 50 3T

ATM = 09

Figure 3 Comparative of ATM implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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Stochastic AnalysisInternational Journal of

4 International Journal of Stochastic Analysis

Black-Scholes term as we do here It is precisely becausevolatility is a deterministic function of the underlying assetprice that we can do that

4 Approximation Formula

In this section we obtain an approximation formula to plainvanilla call price by approximating terms (I)ndash(IV) The mainidea is to use again Lemma 2 to estimate the errors

Theorem6 (BS decomposition formula with error term) Forall 119905 isin [0 119879] one hasE119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120579 (119878119879))] = 119861119878 (119905 119878119905 120579 (119878119905)) + 14

sdot 119903 (1205971198781205792 (119878119905)) 119878119905Γ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2+ 18 (12059721198781205792 (119878119905)) 1205792 (119878119905) 1198782119905 Γ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2+ 124 (1205971198781205792 (119878119905))2 1205792 (119878119905) 1198782119905 Γ2119861119878 (119905 119878119905 120579 (119878119905))sdot (119879 minus 119905)3 + 14 (1205971198781205792 (119878119905)) 1205792 (119878119905)sdot 119878119905ΛΓ119861119878 (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 + Ω

(17)

whereΩ is an error Terms of Ω are written in Appendix A

Proof We apply Lemma 2 to terms (I)ndash(IV) Concretelyfunctions 119860 and 119861 in every case are

(I)

119860(119905 119878119905 1205792 (119878119905)) = (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) 119861119905 = 1199032 int119879

119905(119879 minus 119906) 119889119906 (18)

(II)

119860(119905 119878119905 1205792 (119878119905))= (12059721198781205792 (119878119905)) 1205792 (119878119905) 1198782119905 ΓBS (119905 119878119905 120579 (119878119905))

119861119905 = 14 int119879119905(119879 minus 119906) 119889119906

(19)

(III)

119860(119905 119878119905 1205792 (119878119905))= (1205971198781205792 (119878119905))2 1205792 (119878119905) 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905))

119861119905 = 18 int119879119905(119879 minus 119906)2 119889119906

(20)

(IV)

119860(119905 119878119905 1205792 (119878119905))= (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905 119878119905 120579 (119878119905))

119861119905 = 12 int119879119905(119879 minus 119906) 119889119906

(21)

5 CEV Model

The constant elasticity of variance (CEV) model is a diffusionprocess that solves the stochastic differential equation

119889119878119905 = 119903119878119905119889119905 + 120590119878120573119905 119889119882119905 (22)

Note that writing 120579(119878119905) fl 120590119878120573minus1119905 CEV model can be seenas a local volatility model This model introduced in [5] isone of the first alternatives to Black-Scholes point of view thatappeared in the literature The parameter 120573 ge 0 is called theelasticity of the volatility and 120590 ge 0 is a scale parameter Notethat for 120573 = 1 the model reduces to Osborne-Samuelsonmodel for 120573 = 0 the model reduces to Bachelier model andfor 120573 = 12 the model reduces to Cox-Ingersoll-Ross modelParameter 120573 controls the steepness of the skew exhibited bythe implied volatility

There exists a closed form formula for call options see[5 6] An approximated formula is given in [7]

51 Approximation of the CEV Model Applying Corollary 3to CEV model we obtain the following

Corollary 7 (CEV exact formula) For all 119905 isin [0 119879] one hasE119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120590119878120573minus1119879 )] = 119861119878 (119905 119878119905 120590119878120573minus1119905 )

+ 119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120590119878120573minus1119906 )

sdot (119879 minus 119906) 12059021198782(120573minus1)119905 119889119906] + (120573 minus 1) (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)

sdot 12059041198784(120573minus1)119906 119889119906] + (120573 minus 1)22

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)2

sdot 12059061198786(120573minus1)119906 119889119906] + (120573 minus 1)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ119861119878 (119906 119878119906 120590119878120573minus1119906 ) (119879 minus 119906)

sdot 12059041198784(120573minus1)119906 119889119906]

(23)

We will write

E119905 [119890minus119903(119879minus119905)BS (119879 119878119879 120590119878120573minus1119879 )]= BS (119905 119878119905 120590119878120573minus1119905 ) + (ICEV) + (IICEV) + (IIICEV)

+ (IVCEV) (24)

The exact formula can be difficult to use in practice so wewill use the following approximation

International Journal of Stochastic Analysis 5

Table 1 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 02882882 02882884 minus192119864 minus 07 02882019 864119864 minus 051 10103060 10103070 minus978119864 minus 07 10100377 268119864 minus 0425 24709883 24709894 minus104119864 minus 06 24708310 157119864 minus 045 48771276 48771278 minus222119864 minus 07 48771099 177119864 minus 05

Table 2 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 05356736 05356765 minus289119864 minus 06 05354323 241119864 minus 041 13886303 13886529 minus226119864 minus 05 13868801 175119864 minus 0325 28506826 28507669 minus842119864 minus 05 28450032 568119864 minus 035 51658348 51660433 minus209119864 minus 04 51543092 115119864 minus 02

Table 3 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 13887209 13887438 minus230119864 minus 05 13883284 392119864 minus 041 30389972 30391797 minus183119864 minus 04 30359001 310119864 minus 0325 52954739 52961870 minus713119864 minus 04 52835621 119119864 minus 025 82781049 82800813 minus198119864 minus 03 82459195 322119864 minus 02

Corollary 8 (CEV approximation formula) For all 119905 isin [0 119879]one has

E119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120590119878120573minus1119879 )] = 119861119878 (119905 119878119905 120590119878120573minus1119905 )+ 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2+ 14 (120573 minus 1) (2120573 minus 3) 12059041198784(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)3 + 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + Ω

(25)

whereΩ is an error Terms ofΩ are written in Appendix B Wehave that Ω le (120573 minus 1)2Π(119905 119879 119903 120590 120573) and Π is an increasingfunction on every parameter

Proof Theproof is a direct consequence of applying Lemma 1to (ICEV)ndash(IVCEV) In Appendix C the upper-bounds forevery term are given

52 Numerical Analysis of the Approximation for the CEVCase In this section we compare our numerically approx-imated price of a CEV call option with the following differentpricing methods

(i) The exact formula see [5 6 8] The Matlab code isavailable in [9]

(ii) The Singular Perturbation Technique see [7]

The results for a call option with parameters 120573 = 0251198780 = 100 119870 = 100 120590 = 20 and 119903 = 1 are presented inTable 1

The results in the case that 120573 = 050 are presented inTable 2

The results in the case that 120573 = 075 are presented inTable 3

Finally the results in the case that 120573 = 090 are presentedin Table 4

Note that the new approximation is more accurate thanthe approximation obtained in [7]

In Figure 1 we plot the surface of errors between the exactformula and our approximation

We calculate also the speed time of execution (in seconds)of every method running the function timeit of Matlab 1000times The computer used is an Intel Core i7 CPU Q740

6 International Journal of Stochastic Analysis

Table 4 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 26404164 26404455 minus292119864 minus 05 26401025 314119864 minus 041 55191736 55194053 minus232119864 minus 04 55166821 249119864 minus 0325 91446125 91455159 minus903119864 minus 04 91349142 970119864 minus 035 135553379 135578351 minus250119864 minus 03 135286009 267119864 minus 02

Table 5 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 minus 119905 = 5 and 119903 = 1

Measure Exact formula Approximation HWAverage 256119864 minus 02 173119864 minus 04 167119864 minus 04Standard deviation 303119864 minus 03 286119864 minus 05 252119864 minus 05Max 468119864 minus 02 365119864 minus 04 367119864 minus 04Min 242119864 minus 02 164119864 minus 04 159119864 minus 04

173GHz 173GHz with 4GB of RAM with a Windows 10(times64) The results are presented in Table 5

We observe that singular perturbation method is thefastest method to calculate the price of CEV call option Themethod developed in this work is a little more expensive incomputation time But to compute the exact price is muchmore expensive than any of the other twomethods Note thatin our method we also are able to calculate at the same timethe price and the Gamma of the log-normal price

6 The Approximated Implied VolatilitySurface under CEV Model

In the above section we have computed a bound for theerror between the exact and the approximated pricing for-mulas for the CEV model Now we are going to derive anapproximated implied volatility surface of second order in thelog-moneyness This approximated implied volatility surfacecan help us to understand better the volatility dynamicsMoreover we obtain an approximation of the ATM impliedvolatility dynamics

61 Deriving an Approximated Implied Volatility Surface forthe CEV Model In this section for simplicity and withoutlosing generality we assume 119905 = 0 So 119879 = 120591 denotes timeto maturity The price of an European call option with strike119870 and maturity 119879 is an observable quantity which will bereferred to as 119875obs

0 = 119875obs(119870 119879) The implied volatility isdefined as the value 119868(119879119870) that makes

BS (0 1198780 119868 (119879119870)) = 119875obs0 (26)

Using the results from the previous section we are goingto derive an approximation to the implied volatility as in [10]We use the idea to expand the function with respect to anasymptotic sequence 120575119896infin119896=0 converging to 0 Thus we canwrite

119891 = 1198910 + 1205751198911 + 12057521198912 + 119874 (1205753) (27)

and assuming 120573 isin (0 2) we can choose 120575 = 120573 minus 1 Then wecan expand 119868(119879119870) with respect to this scale as

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870)+ 119874 ((120573 minus 1)3) (28)

and write

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) (29)

Let V0 fl 120590119878120573minus10 Write BS(V0) as a shorthand for BS(01198780 V0)We can rewrite Corollary 8 as

(0 1198780 V0) = BS (V0) + 14 (120573 minus 1)sdot 119879V0 (2119903 + V20 [1 minus 2119889+

V0radic119879])sdot 120597120590BS (V0) + 16 (120573 minus 1)2sdot V30119879 ([1198892+ minus V0radic119879119889+ + 2])sdot 120597120590BS (V0)

(30)

On the other hand we can consider the Taylor expansionof BS(0 1198780 119868(119879119870)) around V0We have that

0 = BS (V0) + 120597120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot) + 12sdot 1205972120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot)2+ sdot sdot sdot

(31)

International Journal of Stochastic Analysis 7

01020304050607080917080K

90100

110120

1

minus1

minus05

15

05

0

130

times10minus4

Figure 1 Error surface between exact formula and our approxima-tion for 1198780 = 100 120590 = 20 and 119903 = 5

and this expression can be rewritten as

BS (119868 (119879119870)) = BS (V0) + (120573 minus 1) 120597120590BS (V0) 1198681 (119879119870)+ (120573 minus 1)2 120597120590BS (V0) 1198682 (119879119870)+ 119874 ((120573 minus 1)2)

(32)

Then equating this expression to 0 we have1198681 (119879119870) = 119879V04 (2119903 + V20 [1 minus 2119889+

V0radic119879])

1198682 (119879119870) = 119879V306 (1198892+ minus V0radic119879119889+ + 2) (33)

Note that 1198681(119879119870) is linear with respect to the log-moneyness while 1198682(119879119870) is quadraticRemark 9 Note that the pricing formula has an error of119874((120573 minus 1)2) as we have proved in Corollary 8 and this istranslated into an error of119874((120573minus1)2) into our approximationof the implied volatility The quadratic term of the volatilityshape is not accurate

We calculate now the short time behavior of the approx-imated implied volatility 119868(119879119870) We write the approximatedequations in terms of 1minus120573 because the case 120573 lt 1 is the mostinteresting and in terms of the log-moneyness ln119870 minus ln 1198780Lemma 10 For 119879 close to 0 one has

119868 (119879119870) asymp V0 minus V02 (1 minus 120573) (ln119870 minus ln 1198780)+ V06 (1 minus 120573)2 (ln119870 minus ln 1198780)2

(34)

Proof Note that

lim119879rarr0

1198681 (119879119870) = V02 (ln119870 minus ln 1198780) lim119879rarr0

1198682 (119879119870) = lim119879rarr0

V301198796 (1198892+ minus V0radic119879119889+ + 2)= V06 (ln119870 minus ln 1198780)2

(35)

Remark 11 Note that (34) is a parabolic equation in the log-moneyness Also from the above expression it is easy to seethat the slope with respect to ln119870 is negative when 119870 lt1198780 exp(32(1 minus120573)) and positive when119870 gt 1198780 exp(32(1 minus120573))showing that the implied volatility for short times tomaturityis smile-shaped This is consistent with the result in [11]Furthermore there is a minimum of the implied volatilitywith respect to ln119870 attained at 119870 = 1198780 exp(32(1 minus 120573))Remark 12 Note that in stochastic volatility models theimplied volatility depends homogeneously on the pair (119878 119870)and in fact it is a function of the log-moneyness ln(1198780119870)As extensively discussed in [12] and exemplified for GARCHoption pricing in [13] this homogeneity property is at oddswith any type of GARCH option pricing We found also thisphenomenon in the quadratic expansion (34)

The behavior of the approximated implied volatility whenthe option is ATM is easy to obtain

119868 (119879119870) = V0 + (V0119903 (120573 minus 1)2 + V30 (120573 minus 1)2

3 )119879

minus (120573 minus 1)2 V6024 1198792(36)

62 Numerical Analysis of the Approximation of the ImpliedVolatility for the CEV Case In this section we com-pare numerically our approximated implied volatilities withimplied volatility computed from call option prices calculatedwith the exact formula and with the ones obtained using thefollowing formula obtained in [7]

119868 (119879119870) = 1205901198911minus120573av

[1 + (1 minus 120573) (2 + 120573)24 (1198650 minus 119870119891av )2

+ (1 minus 120573)224 1205902119879

1198912minus2120573av]

(37)

where 119891av = (12)(1198650 minus 119870) and 1198650 is the forward priceIn Figure 2 we can see that the implied volatility dynam-

ics behaves well for long dated maturities and short datedmaturities when 120573 is close to 1 When this is not the case theformula behaves well at-the-money but the error increases farfrom the ATM value This behavior is a consequence of thequadratic error of our approximation

Comparing the ATM volatility structure we have thefollowing graphics

In Figure 3 we observe that for ATM options theapproximated implied volatility surface fits really well the realimplied volatility structure

Now we put the implied volatility approximation foundin (34) into Black-Scholes formula and compare the obtainedresults with Hagan and Woodward results The results for acall option with parameters 120573 = 025 1198780 = 100 119870 = 100120590 = 20 and 119903 = 1 are presented in Table 6

The results in the case that 120573 = 050 are presented inTable 7

8 International Journal of Stochastic Analysis

0019

00195

002

00205

0021

Impl

ied

vola

tility

10610210098 108 11090 10492 9694K

= 05 T = 1

80 90 100 110 120 13070K

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 1

Exact formulaApproximationHW

92 94 96 98 100 102 104 106 108 11090K

0018

00185

0019

00195

002

00205

0021

00215

Impl

ied

vola

tility

= 05 T = 5

Exact formulaApproximationHW

80 90 100 110 120 13070K

0123

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 5

Figure 2 Comparative of implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

001750018

001850019

00195002

002050021

00215

Impl

ied

vola

tility

05 1 15 4525 3 35 50 2 4T

ATM = 05

Exact formulaApproximation

HWApproximation ATM

012301235

012401245

012501255

012601265

Impl

ied

vola

tility

05 2 41 25 453515 50 3T

ATM = 09

Figure 3 Comparative of ATM implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

Submit your manuscripts athttpswwwhindawicom

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Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 5

Table 1 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 02882882 02882884 minus192119864 minus 07 02882019 864119864 minus 051 10103060 10103070 minus978119864 minus 07 10100377 268119864 minus 0425 24709883 24709894 minus104119864 minus 06 24708310 157119864 minus 045 48771276 48771278 minus222119864 minus 07 48771099 177119864 minus 05

Table 2 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 05356736 05356765 minus289119864 minus 06 05354323 241119864 minus 041 13886303 13886529 minus226119864 minus 05 13868801 175119864 minus 0325 28506826 28507669 minus842119864 minus 05 28450032 568119864 minus 035 51658348 51660433 minus209119864 minus 04 51543092 115119864 minus 02

Table 3 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 13887209 13887438 minus230119864 minus 05 13883284 392119864 minus 041 30389972 30391797 minus183119864 minus 04 30359001 310119864 minus 0325 52954739 52961870 minus713119864 minus 04 52835621 119119864 minus 025 82781049 82800813 minus198119864 minus 03 82459195 322119864 minus 02

Corollary 8 (CEV approximation formula) For all 119905 isin [0 119879]one has

E119905 [119890minus119903(119879minus119905)119861119878 (119879 119878119879 120590119878120573minus1119879 )] = 119861119878 (119905 119878119905 120590119878120573minus1119905 )+ 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2+ 14 (120573 minus 1) (2120573 minus 3) 12059041198784(120573minus1)119905 Γ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)3 + 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓ119861119878 (119905 119878119905 120590119878120573minus1119905 )sdot (119879 minus 119905)2 + Ω

(25)

whereΩ is an error Terms ofΩ are written in Appendix B Wehave that Ω le (120573 minus 1)2Π(119905 119879 119903 120590 120573) and Π is an increasingfunction on every parameter

Proof Theproof is a direct consequence of applying Lemma 1to (ICEV)ndash(IVCEV) In Appendix C the upper-bounds forevery term are given

52 Numerical Analysis of the Approximation for the CEVCase In this section we compare our numerically approx-imated price of a CEV call option with the following differentpricing methods

(i) The exact formula see [5 6 8] The Matlab code isavailable in [9]

(ii) The Singular Perturbation Technique see [7]

The results for a call option with parameters 120573 = 0251198780 = 100 119870 = 100 120590 = 20 and 119903 = 1 are presented inTable 1

The results in the case that 120573 = 050 are presented inTable 2

The results in the case that 120573 = 075 are presented inTable 3

Finally the results in the case that 120573 = 090 are presentedin Table 4

Note that the new approximation is more accurate thanthe approximation obtained in [7]

In Figure 1 we plot the surface of errors between the exactformula and our approximation

We calculate also the speed time of execution (in seconds)of every method running the function timeit of Matlab 1000times The computer used is an Intel Core i7 CPU Q740

6 International Journal of Stochastic Analysis

Table 4 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 26404164 26404455 minus292119864 minus 05 26401025 314119864 minus 041 55191736 55194053 minus232119864 minus 04 55166821 249119864 minus 0325 91446125 91455159 minus903119864 minus 04 91349142 970119864 minus 035 135553379 135578351 minus250119864 minus 03 135286009 267119864 minus 02

Table 5 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 minus 119905 = 5 and 119903 = 1

Measure Exact formula Approximation HWAverage 256119864 minus 02 173119864 minus 04 167119864 minus 04Standard deviation 303119864 minus 03 286119864 minus 05 252119864 minus 05Max 468119864 minus 02 365119864 minus 04 367119864 minus 04Min 242119864 minus 02 164119864 minus 04 159119864 minus 04

173GHz 173GHz with 4GB of RAM with a Windows 10(times64) The results are presented in Table 5

We observe that singular perturbation method is thefastest method to calculate the price of CEV call option Themethod developed in this work is a little more expensive incomputation time But to compute the exact price is muchmore expensive than any of the other twomethods Note thatin our method we also are able to calculate at the same timethe price and the Gamma of the log-normal price

6 The Approximated Implied VolatilitySurface under CEV Model

In the above section we have computed a bound for theerror between the exact and the approximated pricing for-mulas for the CEV model Now we are going to derive anapproximated implied volatility surface of second order in thelog-moneyness This approximated implied volatility surfacecan help us to understand better the volatility dynamicsMoreover we obtain an approximation of the ATM impliedvolatility dynamics

61 Deriving an Approximated Implied Volatility Surface forthe CEV Model In this section for simplicity and withoutlosing generality we assume 119905 = 0 So 119879 = 120591 denotes timeto maturity The price of an European call option with strike119870 and maturity 119879 is an observable quantity which will bereferred to as 119875obs

0 = 119875obs(119870 119879) The implied volatility isdefined as the value 119868(119879119870) that makes

BS (0 1198780 119868 (119879119870)) = 119875obs0 (26)

Using the results from the previous section we are goingto derive an approximation to the implied volatility as in [10]We use the idea to expand the function with respect to anasymptotic sequence 120575119896infin119896=0 converging to 0 Thus we canwrite

119891 = 1198910 + 1205751198911 + 12057521198912 + 119874 (1205753) (27)

and assuming 120573 isin (0 2) we can choose 120575 = 120573 minus 1 Then wecan expand 119868(119879119870) with respect to this scale as

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870)+ 119874 ((120573 minus 1)3) (28)

and write

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) (29)

Let V0 fl 120590119878120573minus10 Write BS(V0) as a shorthand for BS(01198780 V0)We can rewrite Corollary 8 as

(0 1198780 V0) = BS (V0) + 14 (120573 minus 1)sdot 119879V0 (2119903 + V20 [1 minus 2119889+

V0radic119879])sdot 120597120590BS (V0) + 16 (120573 minus 1)2sdot V30119879 ([1198892+ minus V0radic119879119889+ + 2])sdot 120597120590BS (V0)

(30)

On the other hand we can consider the Taylor expansionof BS(0 1198780 119868(119879119870)) around V0We have that

0 = BS (V0) + 120597120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot) + 12sdot 1205972120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot)2+ sdot sdot sdot

(31)

International Journal of Stochastic Analysis 7

01020304050607080917080K

90100

110120

1

minus1

minus05

15

05

0

130

times10minus4

Figure 1 Error surface between exact formula and our approxima-tion for 1198780 = 100 120590 = 20 and 119903 = 5

and this expression can be rewritten as

BS (119868 (119879119870)) = BS (V0) + (120573 minus 1) 120597120590BS (V0) 1198681 (119879119870)+ (120573 minus 1)2 120597120590BS (V0) 1198682 (119879119870)+ 119874 ((120573 minus 1)2)

(32)

Then equating this expression to 0 we have1198681 (119879119870) = 119879V04 (2119903 + V20 [1 minus 2119889+

V0radic119879])

1198682 (119879119870) = 119879V306 (1198892+ minus V0radic119879119889+ + 2) (33)

Note that 1198681(119879119870) is linear with respect to the log-moneyness while 1198682(119879119870) is quadraticRemark 9 Note that the pricing formula has an error of119874((120573 minus 1)2) as we have proved in Corollary 8 and this istranslated into an error of119874((120573minus1)2) into our approximationof the implied volatility The quadratic term of the volatilityshape is not accurate

We calculate now the short time behavior of the approx-imated implied volatility 119868(119879119870) We write the approximatedequations in terms of 1minus120573 because the case 120573 lt 1 is the mostinteresting and in terms of the log-moneyness ln119870 minus ln 1198780Lemma 10 For 119879 close to 0 one has

119868 (119879119870) asymp V0 minus V02 (1 minus 120573) (ln119870 minus ln 1198780)+ V06 (1 minus 120573)2 (ln119870 minus ln 1198780)2

(34)

Proof Note that

lim119879rarr0

1198681 (119879119870) = V02 (ln119870 minus ln 1198780) lim119879rarr0

1198682 (119879119870) = lim119879rarr0

V301198796 (1198892+ minus V0radic119879119889+ + 2)= V06 (ln119870 minus ln 1198780)2

(35)

Remark 11 Note that (34) is a parabolic equation in the log-moneyness Also from the above expression it is easy to seethat the slope with respect to ln119870 is negative when 119870 lt1198780 exp(32(1 minus120573)) and positive when119870 gt 1198780 exp(32(1 minus120573))showing that the implied volatility for short times tomaturityis smile-shaped This is consistent with the result in [11]Furthermore there is a minimum of the implied volatilitywith respect to ln119870 attained at 119870 = 1198780 exp(32(1 minus 120573))Remark 12 Note that in stochastic volatility models theimplied volatility depends homogeneously on the pair (119878 119870)and in fact it is a function of the log-moneyness ln(1198780119870)As extensively discussed in [12] and exemplified for GARCHoption pricing in [13] this homogeneity property is at oddswith any type of GARCH option pricing We found also thisphenomenon in the quadratic expansion (34)

The behavior of the approximated implied volatility whenthe option is ATM is easy to obtain

119868 (119879119870) = V0 + (V0119903 (120573 minus 1)2 + V30 (120573 minus 1)2

3 )119879

minus (120573 minus 1)2 V6024 1198792(36)

62 Numerical Analysis of the Approximation of the ImpliedVolatility for the CEV Case In this section we com-pare numerically our approximated implied volatilities withimplied volatility computed from call option prices calculatedwith the exact formula and with the ones obtained using thefollowing formula obtained in [7]

119868 (119879119870) = 1205901198911minus120573av

[1 + (1 minus 120573) (2 + 120573)24 (1198650 minus 119870119891av )2

+ (1 minus 120573)224 1205902119879

1198912minus2120573av]

(37)

where 119891av = (12)(1198650 minus 119870) and 1198650 is the forward priceIn Figure 2 we can see that the implied volatility dynam-

ics behaves well for long dated maturities and short datedmaturities when 120573 is close to 1 When this is not the case theformula behaves well at-the-money but the error increases farfrom the ATM value This behavior is a consequence of thequadratic error of our approximation

Comparing the ATM volatility structure we have thefollowing graphics

In Figure 3 we observe that for ATM options theapproximated implied volatility surface fits really well the realimplied volatility structure

Now we put the implied volatility approximation foundin (34) into Black-Scholes formula and compare the obtainedresults with Hagan and Woodward results The results for acall option with parameters 120573 = 025 1198780 = 100 119870 = 100120590 = 20 and 119903 = 1 are presented in Table 6

The results in the case that 120573 = 050 are presented inTable 7

8 International Journal of Stochastic Analysis

0019

00195

002

00205

0021

Impl

ied

vola

tility

10610210098 108 11090 10492 9694K

= 05 T = 1

80 90 100 110 120 13070K

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 1

Exact formulaApproximationHW

92 94 96 98 100 102 104 106 108 11090K

0018

00185

0019

00195

002

00205

0021

00215

Impl

ied

vola

tility

= 05 T = 5

Exact formulaApproximationHW

80 90 100 110 120 13070K

0123

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 5

Figure 2 Comparative of implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

001750018

001850019

00195002

002050021

00215

Impl

ied

vola

tility

05 1 15 4525 3 35 50 2 4T

ATM = 05

Exact formulaApproximation

HWApproximation ATM

012301235

012401245

012501255

012601265

Impl

ied

vola

tility

05 2 41 25 453515 50 3T

ATM = 09

Figure 3 Comparative of ATM implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

Submit your manuscripts athttpswwwhindawicom

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Stochastic AnalysisInternational Journal of

6 International Journal of Stochastic Analysis

Table 4 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula Approximation HW119879 minus 119905 Price Price Error Price Error025 26404164 26404455 minus292119864 minus 05 26401025 314119864 minus 041 55191736 55194053 minus232119864 minus 04 55166821 249119864 minus 0325 91446125 91455159 minus903119864 minus 04 91349142 970119864 minus 035 135553379 135578351 minus250119864 minus 03 135286009 267119864 minus 02

Table 5 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 minus 119905 = 5 and 119903 = 1

Measure Exact formula Approximation HWAverage 256119864 minus 02 173119864 minus 04 167119864 minus 04Standard deviation 303119864 minus 03 286119864 minus 05 252119864 minus 05Max 468119864 minus 02 365119864 minus 04 367119864 minus 04Min 242119864 minus 02 164119864 minus 04 159119864 minus 04

173GHz 173GHz with 4GB of RAM with a Windows 10(times64) The results are presented in Table 5

We observe that singular perturbation method is thefastest method to calculate the price of CEV call option Themethod developed in this work is a little more expensive incomputation time But to compute the exact price is muchmore expensive than any of the other twomethods Note thatin our method we also are able to calculate at the same timethe price and the Gamma of the log-normal price

6 The Approximated Implied VolatilitySurface under CEV Model

In the above section we have computed a bound for theerror between the exact and the approximated pricing for-mulas for the CEV model Now we are going to derive anapproximated implied volatility surface of second order in thelog-moneyness This approximated implied volatility surfacecan help us to understand better the volatility dynamicsMoreover we obtain an approximation of the ATM impliedvolatility dynamics

61 Deriving an Approximated Implied Volatility Surface forthe CEV Model In this section for simplicity and withoutlosing generality we assume 119905 = 0 So 119879 = 120591 denotes timeto maturity The price of an European call option with strike119870 and maturity 119879 is an observable quantity which will bereferred to as 119875obs

0 = 119875obs(119870 119879) The implied volatility isdefined as the value 119868(119879119870) that makes

BS (0 1198780 119868 (119879119870)) = 119875obs0 (26)

Using the results from the previous section we are goingto derive an approximation to the implied volatility as in [10]We use the idea to expand the function with respect to anasymptotic sequence 120575119896infin119896=0 converging to 0 Thus we canwrite

119891 = 1198910 + 1205751198911 + 12057521198912 + 119874 (1205753) (27)

and assuming 120573 isin (0 2) we can choose 120575 = 120573 minus 1 Then wecan expand 119868(119879119870) with respect to this scale as

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870)+ 119874 ((120573 minus 1)3) (28)

and write

119868 (119879119870) = V0 + (120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) (29)

Let V0 fl 120590119878120573minus10 Write BS(V0) as a shorthand for BS(01198780 V0)We can rewrite Corollary 8 as

(0 1198780 V0) = BS (V0) + 14 (120573 minus 1)sdot 119879V0 (2119903 + V20 [1 minus 2119889+

V0radic119879])sdot 120597120590BS (V0) + 16 (120573 minus 1)2sdot V30119879 ([1198892+ minus V0radic119879119889+ + 2])sdot 120597120590BS (V0)

(30)

On the other hand we can consider the Taylor expansionof BS(0 1198780 119868(119879119870)) around V0We have that

0 = BS (V0) + 120597120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot) + 12sdot 1205972120590BS (V0)sdot ((120573 minus 1) 1198681 (119879119870) + (120573 minus 1)2 1198682 (119879119870) + sdot sdot sdot)2+ sdot sdot sdot

(31)

International Journal of Stochastic Analysis 7

01020304050607080917080K

90100

110120

1

minus1

minus05

15

05

0

130

times10minus4

Figure 1 Error surface between exact formula and our approxima-tion for 1198780 = 100 120590 = 20 and 119903 = 5

and this expression can be rewritten as

BS (119868 (119879119870)) = BS (V0) + (120573 minus 1) 120597120590BS (V0) 1198681 (119879119870)+ (120573 minus 1)2 120597120590BS (V0) 1198682 (119879119870)+ 119874 ((120573 minus 1)2)

(32)

Then equating this expression to 0 we have1198681 (119879119870) = 119879V04 (2119903 + V20 [1 minus 2119889+

V0radic119879])

1198682 (119879119870) = 119879V306 (1198892+ minus V0radic119879119889+ + 2) (33)

Note that 1198681(119879119870) is linear with respect to the log-moneyness while 1198682(119879119870) is quadraticRemark 9 Note that the pricing formula has an error of119874((120573 minus 1)2) as we have proved in Corollary 8 and this istranslated into an error of119874((120573minus1)2) into our approximationof the implied volatility The quadratic term of the volatilityshape is not accurate

We calculate now the short time behavior of the approx-imated implied volatility 119868(119879119870) We write the approximatedequations in terms of 1minus120573 because the case 120573 lt 1 is the mostinteresting and in terms of the log-moneyness ln119870 minus ln 1198780Lemma 10 For 119879 close to 0 one has

119868 (119879119870) asymp V0 minus V02 (1 minus 120573) (ln119870 minus ln 1198780)+ V06 (1 minus 120573)2 (ln119870 minus ln 1198780)2

(34)

Proof Note that

lim119879rarr0

1198681 (119879119870) = V02 (ln119870 minus ln 1198780) lim119879rarr0

1198682 (119879119870) = lim119879rarr0

V301198796 (1198892+ minus V0radic119879119889+ + 2)= V06 (ln119870 minus ln 1198780)2

(35)

Remark 11 Note that (34) is a parabolic equation in the log-moneyness Also from the above expression it is easy to seethat the slope with respect to ln119870 is negative when 119870 lt1198780 exp(32(1 minus120573)) and positive when119870 gt 1198780 exp(32(1 minus120573))showing that the implied volatility for short times tomaturityis smile-shaped This is consistent with the result in [11]Furthermore there is a minimum of the implied volatilitywith respect to ln119870 attained at 119870 = 1198780 exp(32(1 minus 120573))Remark 12 Note that in stochastic volatility models theimplied volatility depends homogeneously on the pair (119878 119870)and in fact it is a function of the log-moneyness ln(1198780119870)As extensively discussed in [12] and exemplified for GARCHoption pricing in [13] this homogeneity property is at oddswith any type of GARCH option pricing We found also thisphenomenon in the quadratic expansion (34)

The behavior of the approximated implied volatility whenthe option is ATM is easy to obtain

119868 (119879119870) = V0 + (V0119903 (120573 minus 1)2 + V30 (120573 minus 1)2

3 )119879

minus (120573 minus 1)2 V6024 1198792(36)

62 Numerical Analysis of the Approximation of the ImpliedVolatility for the CEV Case In this section we com-pare numerically our approximated implied volatilities withimplied volatility computed from call option prices calculatedwith the exact formula and with the ones obtained using thefollowing formula obtained in [7]

119868 (119879119870) = 1205901198911minus120573av

[1 + (1 minus 120573) (2 + 120573)24 (1198650 minus 119870119891av )2

+ (1 minus 120573)224 1205902119879

1198912minus2120573av]

(37)

where 119891av = (12)(1198650 minus 119870) and 1198650 is the forward priceIn Figure 2 we can see that the implied volatility dynam-

ics behaves well for long dated maturities and short datedmaturities when 120573 is close to 1 When this is not the case theformula behaves well at-the-money but the error increases farfrom the ATM value This behavior is a consequence of thequadratic error of our approximation

Comparing the ATM volatility structure we have thefollowing graphics

In Figure 3 we observe that for ATM options theapproximated implied volatility surface fits really well the realimplied volatility structure

Now we put the implied volatility approximation foundin (34) into Black-Scholes formula and compare the obtainedresults with Hagan and Woodward results The results for acall option with parameters 120573 = 025 1198780 = 100 119870 = 100120590 = 20 and 119903 = 1 are presented in Table 6

The results in the case that 120573 = 050 are presented inTable 7

8 International Journal of Stochastic Analysis

0019

00195

002

00205

0021

Impl

ied

vola

tility

10610210098 108 11090 10492 9694K

= 05 T = 1

80 90 100 110 120 13070K

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 1

Exact formulaApproximationHW

92 94 96 98 100 102 104 106 108 11090K

0018

00185

0019

00195

002

00205

0021

00215

Impl

ied

vola

tility

= 05 T = 5

Exact formulaApproximationHW

80 90 100 110 120 13070K

0123

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 5

Figure 2 Comparative of implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

001750018

001850019

00195002

002050021

00215

Impl

ied

vola

tility

05 1 15 4525 3 35 50 2 4T

ATM = 05

Exact formulaApproximation

HWApproximation ATM

012301235

012401245

012501255

012601265

Impl

ied

vola

tility

05 2 41 25 453515 50 3T

ATM = 09

Figure 3 Comparative of ATM implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 7

01020304050607080917080K

90100

110120

1

minus1

minus05

15

05

0

130

times10minus4

Figure 1 Error surface between exact formula and our approxima-tion for 1198780 = 100 120590 = 20 and 119903 = 5

and this expression can be rewritten as

BS (119868 (119879119870)) = BS (V0) + (120573 minus 1) 120597120590BS (V0) 1198681 (119879119870)+ (120573 minus 1)2 120597120590BS (V0) 1198682 (119879119870)+ 119874 ((120573 minus 1)2)

(32)

Then equating this expression to 0 we have1198681 (119879119870) = 119879V04 (2119903 + V20 [1 minus 2119889+

V0radic119879])

1198682 (119879119870) = 119879V306 (1198892+ minus V0radic119879119889+ + 2) (33)

Note that 1198681(119879119870) is linear with respect to the log-moneyness while 1198682(119879119870) is quadraticRemark 9 Note that the pricing formula has an error of119874((120573 minus 1)2) as we have proved in Corollary 8 and this istranslated into an error of119874((120573minus1)2) into our approximationof the implied volatility The quadratic term of the volatilityshape is not accurate

We calculate now the short time behavior of the approx-imated implied volatility 119868(119879119870) We write the approximatedequations in terms of 1minus120573 because the case 120573 lt 1 is the mostinteresting and in terms of the log-moneyness ln119870 minus ln 1198780Lemma 10 For 119879 close to 0 one has

119868 (119879119870) asymp V0 minus V02 (1 minus 120573) (ln119870 minus ln 1198780)+ V06 (1 minus 120573)2 (ln119870 minus ln 1198780)2

(34)

Proof Note that

lim119879rarr0

1198681 (119879119870) = V02 (ln119870 minus ln 1198780) lim119879rarr0

1198682 (119879119870) = lim119879rarr0

V301198796 (1198892+ minus V0radic119879119889+ + 2)= V06 (ln119870 minus ln 1198780)2

(35)

Remark 11 Note that (34) is a parabolic equation in the log-moneyness Also from the above expression it is easy to seethat the slope with respect to ln119870 is negative when 119870 lt1198780 exp(32(1 minus120573)) and positive when119870 gt 1198780 exp(32(1 minus120573))showing that the implied volatility for short times tomaturityis smile-shaped This is consistent with the result in [11]Furthermore there is a minimum of the implied volatilitywith respect to ln119870 attained at 119870 = 1198780 exp(32(1 minus 120573))Remark 12 Note that in stochastic volatility models theimplied volatility depends homogeneously on the pair (119878 119870)and in fact it is a function of the log-moneyness ln(1198780119870)As extensively discussed in [12] and exemplified for GARCHoption pricing in [13] this homogeneity property is at oddswith any type of GARCH option pricing We found also thisphenomenon in the quadratic expansion (34)

The behavior of the approximated implied volatility whenthe option is ATM is easy to obtain

119868 (119879119870) = V0 + (V0119903 (120573 minus 1)2 + V30 (120573 minus 1)2

3 )119879

minus (120573 minus 1)2 V6024 1198792(36)

62 Numerical Analysis of the Approximation of the ImpliedVolatility for the CEV Case In this section we com-pare numerically our approximated implied volatilities withimplied volatility computed from call option prices calculatedwith the exact formula and with the ones obtained using thefollowing formula obtained in [7]

119868 (119879119870) = 1205901198911minus120573av

[1 + (1 minus 120573) (2 + 120573)24 (1198650 minus 119870119891av )2

+ (1 minus 120573)224 1205902119879

1198912minus2120573av]

(37)

where 119891av = (12)(1198650 minus 119870) and 1198650 is the forward priceIn Figure 2 we can see that the implied volatility dynam-

ics behaves well for long dated maturities and short datedmaturities when 120573 is close to 1 When this is not the case theformula behaves well at-the-money but the error increases farfrom the ATM value This behavior is a consequence of thequadratic error of our approximation

Comparing the ATM volatility structure we have thefollowing graphics

In Figure 3 we observe that for ATM options theapproximated implied volatility surface fits really well the realimplied volatility structure

Now we put the implied volatility approximation foundin (34) into Black-Scholes formula and compare the obtainedresults with Hagan and Woodward results The results for acall option with parameters 120573 = 025 1198780 = 100 119870 = 100120590 = 20 and 119903 = 1 are presented in Table 6

The results in the case that 120573 = 050 are presented inTable 7

8 International Journal of Stochastic Analysis

0019

00195

002

00205

0021

Impl

ied

vola

tility

10610210098 108 11090 10492 9694K

= 05 T = 1

80 90 100 110 120 13070K

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 1

Exact formulaApproximationHW

92 94 96 98 100 102 104 106 108 11090K

0018

00185

0019

00195

002

00205

0021

00215

Impl

ied

vola

tility

= 05 T = 5

Exact formulaApproximationHW

80 90 100 110 120 13070K

0123

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 5

Figure 2 Comparative of implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

001750018

001850019

00195002

002050021

00215

Impl

ied

vola

tility

05 1 15 4525 3 35 50 2 4T

ATM = 05

Exact formulaApproximation

HWApproximation ATM

012301235

012401245

012501255

012601265

Impl

ied

vola

tility

05 2 41 25 453515 50 3T

ATM = 09

Figure 3 Comparative of ATM implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Stochastic Analysis

0019

00195

002

00205

0021

Impl

ied

vola

tility

10610210098 108 11090 10492 9694K

= 05 T = 1

80 90 100 110 120 13070K

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 1

Exact formulaApproximationHW

92 94 96 98 100 102 104 106 108 11090K

0018

00185

0019

00195

002

00205

0021

00215

Impl

ied

vola

tility

= 05 T = 5

Exact formulaApproximationHW

80 90 100 110 120 13070K

0123

0124

0125

0126

0127

0128

0129

Impl

ied

vola

tility

= 09 T = 5

Figure 2 Comparative of implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

001750018

001850019

00195002

002050021

00215

Impl

ied

vola

tility

05 1 15 4525 3 35 50 2 4T

ATM = 05

Exact formulaApproximation

HWApproximation ATM

012301235

012401245

012501255

012601265

Impl

ied

vola

tility

05 2 41 25 453515 50 3T

ATM = 09

Figure 3 Comparative of ATM implied volatility approximations for 1198780 = 100 120590 = 20 and 119903 = 5

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 9

Table 6 Call option 120573 = 025 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 02882882 02882882 351119864 minus 08 028820185 864119864 minus 051 10103060 10103057 236119864 minus 07 1010037675 268119864 minus 0425 24709883 24709880 294119864 minus 07 2470830954 157119864 minus 045 48771276 48771275 672119864 minus 08 4877109923 177119864 minus 05

Table 7 Call option 120573 = 050 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 05356736 05356732 427119864 minus 07 053543231 241119864 minus 041 13886303 13886267 365119864 minus 06 1386880117 175119864 minus 0325 28506826 28506672 154119864 minus 05 2845003212 568119864 minus 035 51658348 51657911 436119864 minus 05 5154309238 115119864 minus 02

The results in the case that 120573 = 075 are presented inTable 8

And the results in the case that 120573 = 090 are presented inTable 9

Our approximation is better than Hagan and Woodwardone

We compare also execution times (see Table 10)We can observe that both formulas are similar in compu-

tation time with the new approximation formula being a bitfaster

63 Calibration of the Model Following the ideas of [2] wepropose a method to calibrate the model This method willallow us to find 120590 and 120573 using quadratic linear regressionWe can recover the parameters with a set of options of thesame maturity with (34) or with ATM options of differentmaturities (36)

631 Calibration Using the Smile of Volatility Using a set ofoptions with the same maturity and the parameters 1198780 = 100120590 = 20 119903 = 5 119870 = 98 sdot sdot sdot 102 119879 = 1 and 120573 = 05 Wecalculate the price and their implied volatilities with the exactformula We do a quadratic regression adjusting a parabola119886 + 119887119888 + 1198881199092 with 119909 = ln119870 minus ln 1198780 to the implied volatilitiesUsing (34) it is easy to see that 120573 = 2119887119886 + 1 and 120590 = 119886119878120573minus1In this case we have

00002004461199092 minus 000497683119909 + 0020000611 (38)

from which we obtain 120573 = 050233 and 120590 = 19787Using the same procedure for 119879 = 5 and 120573 = 05 we find

that

minus00012343081199092 minus 0004881387119909 + 0020013633 (39)

from which we obtain 120573 = 051219 and 120590 = 18921

Using the same procedure for 119879 = 1 and 120573 = 09 we findthat

00003828761199092 minus 0006311173119909 + 0126192162 (40)

from which we obtain 120573 = 089997 and 120590 = 20002Using the same procedure for 119879 = 5 and 120573 = 09 we find

that

0000103931199092 minus 000628411119909 + 0126198861 (41)

from which we obtain 120573 = 090041 and 120590 = 19963

632 Calibration Using ATM Implied Volatilities Using aset of ATM options with the same maturity and parameters1198780 = 100 120590 = 20 119903 = 5 119879 = 03 05 08 09 1 and120573 = 05 we calculate the price and their implied volatilitieswith the exact formula Then we do a quadratic regressionadjusting a parabola 119886 + 119887119888 + 1198881199092 with 119909 = 119879 to the impliedvolatilities Using (36) it is easy to see that 120573 = 1 + (minus3119903 plusmnradic91199032 + 16119886119887)41198862 and 120590 = 119886119878120573minus1 In this case we have

000000861199092 minus 00002577119909 + 00200020 (42)

from which we obtain 120573 = 048324 (or 120573 = minus18594 whichwe can discard) and 120590 = 21607

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 05we find that

000000241199092 minus 00002495119909 + 00199997 (43)

from which we obtain 120573 = 049970 (or 120573 = minus1860055 whichwe can discard) and 120590 = 20028

Using the same procedure for 119879 = 03 05 08 09 1 and120573 = 09 we find that

minus000000541199092 minus 00003076119909 + 01261899 (44)

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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Stochastic AnalysisInternational Journal of

10 International Journal of Stochastic Analysis

Table 8 Call option 120573 = 075 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 13887209 13887176 329119864 minus 06 1388328423 392119864 minus 041 30389972 30389707 264119864 minus 05 3035900123 310119864 minus 0325 52954739 52953686 105119864 minus 04 5283562121 119119864 minus 025 82781049 82778040 301119864 minus 04 8245919491 322119864 minus 02

Table 9 Call option 120573 = 090 1198780 = 100 119870 = 100 120590 = 20 and 119903 = 1

Parameters Exact formula BS with implied volatility (34) HW119879 Price Price Error Price Error025 26404164 26404122 417119864 minus 06 2640102524 314119864 minus 041 55191736 55191404 331119864 minus 05 551668209 249119864 minus 0325 91446125 91444830 129119864 minus 04 9134914233 970119864 minus 035 135553379 135549787 359119864 minus 04 1352860091 267119864 minus 02

Table 10 Call option 120573 = 09 1198780 = 100 119870 = 100 120590 = 20 119879 = 5 and 119903 = 1

Measure HW BS with implied volatility (34)Average 167119864 minus 04 166119864 minus 04Standard deviation 252119864 minus 05 237119864 minus 05Max 367119864 minus 04 348119864 minus 04Min 159119864 minus 04 158119864 minus 04

from which we obtain 120573 = 090040 (or 120573 = minus36103 whichwe can discard) and 120590 = 19963

Using the same procedure for119879 = 1 2 3 4 5 and120573 = 09we find that

000000061199092 minus 00003141119909 + 01261907 (45)

from which we obtain 120573 = 089822 (or 120573 = minus36081 whichwe can discard) and 120590 = 20164

We have seen that to do a quadratic regression is enoughto recover a good approximation of the parameters

7 Conclusion

In this paper we notice that ideas developed in [1] for Hestonmodel can be used for spot-dependent volatility models Itis interesting to realize that the approximation found in thiscase has more terms than the one obtained for stochasticvolatility models (see [4]) We have applied this technique tothe CEV model doing a comparison between exact pricesBlack-Scholes using Hagan andWoodward implied volatilityand our price approximation We have seen that our approx-imation is better than Hagan and Woodward approximationfor pricing but a bit more expensive in computation timeAs well we have calculated an approximation of the impliedvolatility as the limit of the implied volatility close to zeroas a function of log-moneyness and an approximation ofthe ATM implied volatility as a function of time We havecompared our approximationwith the exact implied volatility

and Hagan and Woodward approximation We note thatif we put our implied volatility approximation into Black-Scholes function we get a better approximation than Haganand Woodward in the same computation time So we havedeveloped an easy way to calibrate CEV model that consistsessentially in doing a quadratic regression

Appendix

In the following appendices we obtain the error terms of thedecomposition inTheorem 6 (Appendix A) the same formu-las in the particular case of CEV model (Appendix B) andupper-bounds for those terms using Lemma 1 (Appendix C)In all the section we write 120591119906 fl 119879 minus 119906A Decomposition Formulas in the

General Model

A1 Decomposition of Term (I) The term (I) can be decom-posed by

1199032E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (S119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 119878119906119889119906] minus 1199034 (1205971198781205792 (119878119905)) 119878119905ΓBS (119905 119878119905 120579 (119878119905)) (119879minus 119905)2 = 11990328 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 11

sdot 1205913119906 (1205971198781205792 (119878119906))2 1198782119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 11990332sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A1)

A2 Decomposition of Term (II) The term (II) can be decom-posed by

14E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906119889119906] minus 18 (12059721198781205792 (119878119905)) 1205792 (119878119905)sdot 1198782119905 ΓBS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059721198781205792 (119878119906) ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990316sdot E119905 [int119879

119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) 1205792 (119878119906)

sdot 1198783119906Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 119889119906]+ 116E119905 [int

119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906))

sdot (1205971198781205792 (119878119906))2 1205792 (119878u) 1198784119906119889119906] + 164

sdot E119905 [int119879119905119890minus119903(119906minus119905) (12059721198781205792 (119878119906)) Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))2 1205794 (119878119906) 1198784119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059721198781205792 (119878119906)

sdot 1198782119906ΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 116E119905 [int

T

119905119890minus119903(119906minus119905)Λ((12059721198781205792 (119878119906))

sdot 1205792 (119878119906) 1198782119906Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906))sdot 1205792 (119878119906) 119878119906119889119906]

(A2)

A3 Decomposition of Term (III) The term (III) can bedecomposed by

18E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906))2

sdot 1205792 (119878119906) 1198782119906119889119906] minus 124 (1205971198781205792 (119878119905))2 1205792 (119878119905)sdot 1198782119905 Γ2BS (119905 119878119905 120579 (119878119905)) (119879 minus 119905)3 = 11990324sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 11990348sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906)) 1198783119906119889119906] + 148sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 Γ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906] + 196sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198784119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 1205914119906 (1205971198781205792 (119878119906))4 1205792 (119878119906) 1198784119906119889119906] + 1192sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))2 1205792 (119878119906)

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

12 International Journal of Stochastic Analysis

sdot Γ4BS (119906 119878119906 120579 (119878119906)) 1205915119906 (1205971198781205792 (119878119906))2 1205792 (119878119906) 1198784119906119889119906]+ 124E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2

sdot Γ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906]+ 148E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot Γ3BS (119906 119878119906 120579 (119878119906))) 1205914119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906]

(A3)

A4 Decomposition of Term (IV) The term (IV) can bedecomposed by

12E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120579 (119878119906)) 120591119906 (1205971198781205792 (119878119906))

sdot 1205792 (119878119906) 119878119906119889119906] minus 14 (1205971198781205792 (119878119905)) 1205792 (119878119905) 119878119905ΛΓBS (119905119878119905 120579 (119878119905)) (119879 minus 119905)2 = 1199034E119905 [int

119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906))

sdot ΛΓBS (119906 119878119906 120579 (119878119906)) 1205912119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 1199038sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (1205971198781205792 (119878119906)) 1198782119906119889119906] + 18sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) ΛΓBS (119906 119878119906 120579 (119878119906))

sdot 1205912119906 (12059721198781205792 (119878119906)) 1205792 (119878119906) 1198783119906119889119906] + 116sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ2BS (119906 119878119906 120579 (119878119906)) 1205913119906 (12059721198781205792 (119878119906)) 1205792 (119878119906)sdot 1198783119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 1205913119906 (1205971198781205792 (119878119906))3 1205792 (119878119906) 1198783119906119889119906] + 132sdot E119905 [int119879

119905119890minus119903(119906minus119905) (1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot ΛΓ3BS (119906 119878119906 120579 (119878119906)) 1205914119906 (1205971198781205792 (119878119906))2 1205792 (119878119906)

sdot 1198783119906119889119906] + 14E119905 [int119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906))

sdot 119878119906ΛΓBS (119906 119878119906 120579 (119878119906))) 1205912119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906] + 18E119905 [int

119879

119905119890minus119903(119906minus119905)Λ((1205971198781205792 (119878119906)) 1205792 (119878119906)

sdot 119878119906ΛΓ2BS (119906 119878119906 120579 (119878119906))) 1205913119906 (1205971198781205792 (119878119906)) 1205792 (119878119906)sdot 119878119906119889119906]

(A4)

B Decomposition Formulas for theCEV Model

B1 Decomposition of the Term (I119862119864119881)

119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059111990612059021198782(120573minus1)119906 119889119906] minus 12 (120573 minus 1)sdot 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 11990322 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059041198784(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199034 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906]

(B1)

B2 Decomposition of the Term (II119862119864119881)

12 (120573 minus 1) (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2 = 1199032 (120573 minus 1)2

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 13

sdot (2120573 minus 3)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199034 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)12059061198786(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 1205913119906119889119906]

+ 14 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

+ 18 (120573 minus 1)2 (2120573 minus 3)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120579 (119878119906)) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 18 (2120573 minus 3) (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΓBS (119906 S119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B2)

B3 Decomposition of the Term (III119862119864119881)

12 (120573 minus 1)2 E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119879 )

sdot 120591211990612059061198786(120573minus1)119906 119889119906] minus 16 (120573 minus 1)2sdot 12059061198786(120573minus1)119906 Γ2BS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3 = 1199033 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 1199036 (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591411990612059081198788(120573minus1)119906 119889119906] + 16 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ 112 (120573 minus 1)3 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)4 E119905 [int119879119905119890minus119903(119906minus119905)Γ3BS (119906 119878119906 120579 (119878119906))

sdot 12059141199061205901011987810(120573minus1)119906 119889119906] + 112 (120573 minus 1)4

sdot E119905 [int119879119905119890minus119903(119906minus119905)Γ4BS (119906 119878119906 120590119878120573minus1119906 ) 12059151199061205901211987812(120573minus1)119906 119889119906]

+ 13 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059041198784120573minus6119906 Γ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 1205913119906 (12059041198784(120573minus1)119906 ) 119889119906] + 16 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)1198782119906Λ(12059061198786120573minus8119906 Γ3BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591411990612059041198784(120573minus1)119906 119889119906] (B3)

B4 Decomposition of the Term (IV119862119864119881)

(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 12059041198784(120573minus1)119906 120591119906119889119906]

minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2= 119903 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 1199032 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059061198786(120573minus1)119906 119889119906]

+ 12 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 ) 120591211990612059061198786(120573minus1)119906 119889119906]

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

14 International Journal of Stochastic Analysis

+ 14 (120573 minus 1)2 (2120573 minus 3)sdot E119905 [int119879

119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ) 120591311990612059081198788(120573minus1)119906 119889119906]

+ (120573 minus 1)3 E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ2BS (119906 119878119906 120579 (119878119906))

sdot 120591311990612059081198788(120573minus1)119906 119889119906] + 14 (120573 minus 1)3

sdot E119905 [int119879119905119890minus119903(119906minus119905)ΛΓ3BS (119906 119878119906 120590119878120573minus1119906 ) 12059141199061205901011987810(120573minus1)119906 119889119906]

+ (120573 minus 1)2sdot E119905 [int119879

119905119890minus119903(119906minus119905)Λ(12059021198782(120573minus1)119906 ΛΓBS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591211990612059041198784(120573minus1)119906 119889119906] + 12 (120573 minus 1)2

sdot E119905 [int119879119905119890minus119903(119906minus119905)Λ(12059041198784(120573minus1)119906 ΛΓ2BS (119906 119878119906 120590119878120573minus1119906 ))

sdot 120591311990612059041198784(120573minus1)119906 119889119906] (B4)

C Upper-Bound First-OrderDecomposition Formulas

C1 Upper-Bound of the Term (I119862119864119881)

100381610038161003816100381610038161003816100381610038161003816119903 (120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059021198782(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 11990312059021198782(120573minus1)119905 ΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990322 1198621 (120573 minus 1)2

sdot 120590 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573 minus 1)2 (2120573 minus 3)

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198623 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198624119903 (120573 minus 1)3 1205903

sdot int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990321198625 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2Π1 (119905 119903 120590 120573)

(C1)

where Π1(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 5) are some constants and 119862 =max(119862119894)

C2 Upper-Bound of the Term (II119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1) (2120573 minus 3)E119905 [int119879

119905119890minus119903(119906minus119905)ΓBS (119906 119878119906 120590119878120573minus1119879 )

sdot 12059111990612059041198784(120573minus1)119906 119889119906] minus 14 (120573 minus 1) (2120573 minus 3)sdot 12059041198784(120573minus1)119905 ΓBS (119905 119878119905 120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le

11990321198621 (120573minus 1)2 (2120573 minus 3) 1205903 int119879

119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990341198622 (120573

minus 1)2 (2120573 minus 3) 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 141198623 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198624 (120573

minus 1)2 (2120573 minus 3)2 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198625 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 181198626 (2120573

minus 3) (120573 minus 1)3 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1198627 (120573

minus 1)3 (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 121198628 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198629 (120573 minus 1)3

sdot (2120573 minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1411986210 (120573

minus 1)2 (2120573 minus 3) 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)2

sdot Π2 (119905 119903 120590 120573)

(C2)

where Π2(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C3 Upper-Bound of the Term (III119862119864119881)

10038161003816100381610038161003816100381610038161003816100381612 (120573 minus 1)2 E119905 [int119879

119905119890minus119903(119906minus119905)Γ2BS (119906 S119906 120590119878120573minus1119879 )

sdot 12059061198786(120573minus1)119906 1205912119906119889119906] minus 16 (120573 minus 1)2 12059061198786(120573minus1)119906 Γ2BS (119905119878119905 120590119878120573minus1119905 ) (119879 minus 119905)3100381610038161003816100381610038161003816100381610038161003816 le

11990331198621 (120573 minus 1)3

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 11990361198622 (120573 minus 1)3

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 15

sdot 1205903 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 161198623 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198624 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198625 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1121198626 (120573 minus 1)4

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 231198627 (120573 minus 1)3 (2120573

minus 3) 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 131198628 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 131198629 (120573 minus 1)3 (3120573 minus 4)

sdot 1205905 int119879119905119890minus119903(119906minus119905) (radic120591119906)3 119889119906 + 1611986210 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 le 119862 (120573 minus 1)3Π3 (119905 119903 120590 120573)

(C3)

where Π3(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)C4 Decomposition of the Term (IV119862119864119881)

100381610038161003816100381610038161003816100381610038161003816(120573 minus 1)E119905 [int119879119905119890minus119903(119906minus119905)ΛΓBS (119906 119878119906 120590119878120573minus1119906 )

sdot 12059041198784(120573minus1)119906 120591119906119889119906] minus 12 (120573 minus 1) 12059041198784(120573minus1)119905 ΛΓBS (119905 119878119905120590119878120573minus1119905 ) (119879 minus 119905)2100381610038161003816100381610038161003816100381610038161003816 le 1198621119903 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 11990321198622 (120573 minus 1)2

sdot 1205902 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 121198623 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198624 (120573 minus 1)2 (2120573 minus 3)

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198625 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 141198626 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 21198627 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1198628 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 + 21198629 (120573 minus 1)3

sdot 1205904 int119879119905119890minus119903(119906minus119905)120591119906119889119906 + 1211986210 (120573 minus 1)2

sdot 1205903 int119879119905119890minus119903(119906minus119905)radic120591119906119889119906 le 119862 (120573 minus 1)2Π4 (119905 119903 120590 120573)

(C4)

where Π4(119905 119879 119903 120590 120573) is an increasing function for everyparameter 119862119894 (119894 = 1 10) are some constants and 119862 =max(119862119894)Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Josep Vives was partially supported by Grant MEC MTM2013 40782 P

References

[1] E Alos ldquoA decomposition formula for option prices in the Hes-ton model and applications to option pricing approximationrdquoFinance and Stochastics vol 16 no 3 pp 403ndash422 2012

[2] E Alos R de Santiago and J Vives ldquoCalibration of stochasticvolatility models via second-order approximation the Hestoncaserdquo International Journal of Theoretical and Applied Financevol 18 no 6 Article ID 1550036 1550036 31 pages 2015

[3] E Alos J A Leon and J Vives ldquoOn the short-time behavior ofthe implied volatility for jump-diffusionmodels with stochasticvolatilityrdquo Finance and Stochastics vol 11 no 4 pp 571ndash5892007

[4] R Merino and J Vives ldquoA generic decomposition formulafor pricing vanilla options under stochastic volatility modelsrdquoInternational Journal of Stochastic Analysis Article ID 103647Art ID 103647 11 pages 2015

[5] J C Cox ldquoThe constant elasticity of variance option pricingmodelrdquo Journal of PortfolioManagement vol 22 pp 16-17 1996

[6] D C Emanuel and J D MacBeth ldquoFurther results on theconstant elasticity of variance call option pricing modelrdquo TheJournal of Financial and Quantitative Analysis vol 17 no 4 pp533ndash554 1982

[7] P S Hagan and D EWoodward ldquoEquivalent Black volatilitiesrdquoApplied Mathematical Finance vol 6 no 3 pp 147ndash157 2010

[8] M SCHRODER ldquoComputing the Constant Elasticity of Vari-ance Option Pricing Formulardquo The Journal of Finance vol 44no 1 pp 211ndash219 1989

[9] J Kienitz and D Wetterau ldquoFinancial Modelling TheoryImplementation and Practice (with Matlab Source)rdquo FinancialModelling Theory Implementation and Practice (with MatlabSource) pp 1ndash719 2013

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 International Journal of Stochastic Analysis

[10] J-P Fouque G Papanicolaou R Sircar and K Soslashlna ldquoSingularperturbations in option pricingrdquo SIAM Journal on AppliedMathematics vol 63 no 5 pp 1648ndash1665 2003

[11] E Renault and N Touzi ldquoOption hedging and implied volatili-ties in a stochastic volatility modelrdquoMathematical Finance vol6 no 3 pp 279ndash302 1996

[12] D M Kreps and K F Wallis Advances in Economics andEconometrics Theory and Applications Cambridge UniversityPress Cambridge 1997

[13] R Garcia and E Renault ldquoA note on garch option pricing andhedgingrdquoMathematical Finance vol 8 no 2 pp 153ndash161 1998

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of