on the impact of correlation on option prices: a malliavin ...calculate the malliavin derivative of...

On the impact of correlation on option prices: a Malliavin Calculus approach

E. Alòs (2006): A generalization of the Hull and White formula with applicationsto option pricing approximation. Finance and Stochastics 10 (3), 353-365.

E. Alòs, J. A. León and J. Vives (2007): On the short-time behaviour of theimplied volatility for stochastic volatility models with jumps. Finance and Stochastics 11 (4), 571-589

RESULTS FROM

STOCHASTIC VOLATILITY MODELS

Stochastic volatility models allow us to describe the smiles andskews observed in real market data:

( )*2*2 12

1ttttt dBdWdtrdX ρρσσ −++

−=

0=ρ

Log-price Volatility (stochastic, adapted to the filtrationgenerated by W)

Implied volatility smile Implied volatility skew 0≠ρ

SOME QUESTIONS AND MOTIVATION

How to quantify the impact of correlation on option prices?

What about the term structure?

Option price=option price in the uncorrelated case

(classical Hull and White formula)+ correction due by correlation

We will develop a formula of the form

This result will allow us to describe the impact of the correlation on theoption prices. As an application, we can use it to construct option pricing

approximation formulas, or to study the short-time behaviour of the impliedvolatility for stochastic volatility models with jumps.

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (I)

Calculate the Malliavin derivative of a diffusion process

Use the duality relationship between the Malliavinderivative and the Skorohod integral to develop adequate

change-of-variable formulas for anticipating processes

Here our pourpose is to present the basic concepts onMalliavin calculus that have been used up to now in financial

applications. Basically, we will see how to:

MAIN IDEA: THE FUTURE INTEGRATED VOLATILITY IS AN ANTICIPATING PROCESS

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (II)

Malliavin derivative : definition

( ) ( ) ( )( )[ ]( )Ω×

=

TL

hWhWhWfF n

,0in variablerandom

...,,2

21

( ) [ ]( ) processGaussian

,0, 2 TLhhW ∈

( ) ( ) ( )( ) ( )

[ ]( )Ω×

∂∂=∑

TL

thhWhWhWx

fFD in

it

,0in DerivativeMalliavin

...,,

2

21

(closable operator)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (III)

Malliavin derivative: examples

[ ] )(1

Scholes)-(Black 2

-rexp

,0

2

t

rSSD

WtS

tttW

r

t

σ

σσ

=

+

=

[ ] )(1 ,0 tWD ssW

t =

( ) ( )

( )[ ] )(1

)(

,0

00

rceYD

UhlenbeckOrnstein

dWecemYmY

trt

tW

r

r

t rttt

−−

−−−

=

−

+−+= ∫

α

αα

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (IV)

Skorohod integral: definition

It is the adjoint of the Malliavin derivative operator:

[ ]( ) ( ) ( )hWhTLh W =⇒∈ δ,02

( )( ) ( ) SFdsuFDEFuE s

T Ws

W ∈= ∫ allfor ,0

δ

Example:

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (V)

Skorohod integral: properties

The Skorohod integral of a process multiplied by a random variable

( )∫∫∫ +=T

sWs

T

ss

T

ss dsuFDdWuFdWFu000

The Skorohod integral is an extension ofthe classical Itô integral

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI)

THE ANTICIPATING ITÔ’S FORMULA

( ) ( ) ( )

( ) ( )( ) ,''2

1'

'

00

00

∫∫

∫

∇++

+=

t

sss

t

ss

t

ssst

dsuuXFdsvXF

dWuXFXFXF

∫ ∫++=t t

ssst dsvdWuXX0 00

Non necessarily adapted

( ) drvDdWuDuus

rWs

s

rrWsss ∫∫ ++=∇

0022 : where

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI)

Proof (sketch)

.0 assume wesimplicity of sake For the ≡v

( ) ( ) ( )

( )∑ ∫

∑ ∫

+

+=

+

+

2

0

1

1

''2

1

'

i

ii

i

ii

t

t sst

t

t sstt

dWuXF

dWuXFXFXF

We proceed as in the proof of the classical Itô’s formula

∫t

s dsu0

2

2

1

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VII)

( )( ) ( )∑∫∑∫

∑ ∫++

+

−=

11

1

''

'

i

ii

i

ii

i

ii

t

t stWs

t

t sst

t

t sst

dsuXFDdWuXF

dWuXF

∫t

sss dWuXF0

)('2

1

∫ ∫

t

s

s

rrWss dsudWuDXF

0 0)(''

( )[ ]∫−t

ssWs dsuXFD

0'

AN EXTENSION OF THE HULL AND WHITE FORMULA (I)

[ ]ttTr

t FHeEV )(* −−=

option price payoff

( )TTT XTBSV ν;,=

The Black-Scholes function final conditionimplies that

−= ∫ ds

tTv

T

t

st22 1 σ

Basic idea

Black-Scholespricing formula

Log-price

where

AN EXTENSION OF THE HULL AND WHITE FORMULA (II)

Then

( )( )[ ]tTT

tTtTr

t

FXTBSE

FVEeV

ν;,*

*)(

=

= −−

The classical Hull and White term

( )( )ttt FXtBSE ν;,*

is the option price in the uncorrelated case

com

pare

Then we want to evaluate the difference

( ) ( )ttTT XtBSXTBS νν ;,;, −

We need to construct an adequateanticipating Itô’s formula

process nganticipatian is tν

AN EXTENSION OF THE HULL AND WHITE FORMULA (III)

AN EXTENSION OF THE HULL AND WHITE FORMULA (IV)

Anticipating Itô’s formula

( ) ( ) ( )

( ) ( )

( )( )

( )∫

∫

∫ ∫

∫

∂∂+

∂∂∂+

∂∂+

∂∂+

∂∂+=

−

t

sss

t

ssss

t t

ssssss

t

sstt

dsuYXsx

F

dsuYDYXsyx

F

dYYXsy

FdXYXs

x

F

dsYXss

FYXFYXtF

0

22

2

0

2

0 0

000

,,

,,

,,,,

,,,,0,,

∫=T

t st dsY θ ( ) ∫=− T

s rWss drDYD θ:

additionalterm

AN EXTENSION OF THE HULL AND WHITE FORMULA (V)

( )

−= −−

ttrt

ttrt Y

tTXtBSeXtBSe

1;,;, ν

Main result:

the extension of the Hull and White formula

We apply the above Itô’s anticipating formula to the process

and we obtain

AN EXTENSION OF THE HULL AND WHITE FORMULA (VI)

( ) ( )

( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( )∫

∫

∫

∫

−−

∂∂−

−∂∂∂+

−+∂

∂+

∂∂−

∂∂−++

==

−

−−

−

−

−−−

T

ts

ssss

rs

t

sss

ssrs

tss

T

t ssrs

T

t sssssBSrs

ttrt

TTrT

TrT

dssT

XsBS

e

dsYDsT

Xsx

BSe

dZdWXsx

BSe

dsXsBSxx

Le

XtBSeXTBSeVe

ννσν

σ

σν

νσ

ρ

ρρσν

ννσν

νν

22

0

2

*2*

2

222

,,2

1

1,,

2

1,,

,,2

1

,,,,

Black-Scholes differential operator

Cancel

Zero expectation

AN EXTENSION OF THE HULL AND WHITE FORMULA (VII)

( ) ( )( )( ) ( )∫

−−

−−

−∂∂∂+

=t

sss

ssrs

tttrt

tTrT

dsYDsT

Xsx

BSe

FXtBSEeFVEe

0

2

**

)(

1,,

2

,,

σν

νσ

ρ

ν

( ) ( )ssss XsHXsxx

νν ,,:,,2

2

3

3

=

∂∂−

∂∂

s

T

s rWss drD σσ

=Λ ∫

2*

:

( ) ( )( )( )

( ) ( )

Λ+

=

=

∫−−

−−

t

t

ssstsr

ttt

tTtTr

t

FdsXsHeE

FXtBSE

FVEeV

0

*

*

*

,,2

,,

νρν

AN EXTENSION OF THE HULL AND WHITE FORMULA (VIII)

( ) ( )

Λ∫

−−t

t

ssstsr FdsXsHeE

0

* ,,2

νρ

The above arguments do not requiere the volatility to be Markovian.

The main contribution of this formula is to describe the effect of thecorrelation as the term

APPLICATIONS TO OPTION PRICING APPROXIMATION (I)

( )( )

( )∫

∫

−=

Λ+

=

T

t tst

t

T

t stt

ttaprox

dsFEtT

FdsEXtH

XtBSV

2**

**

*

1

,,,2

,,

σν

νρν

Consider the approximation

APPLICATIONS TO OPTION PRICING APPROXIMATION (II)

( )( )

enoughregular and

with ,

f

dWdtYmdY

Yf

ttt

ss

αλα

σ

+−=

=

Using Malliavin calculus again we can see that, in the case

( )ααλ

ln12

+≤− CVV aproxt

A similar result was proven by Alòs and Ewald(2008) for the Heston volatility model

APPLICATIONS TO OPTION PRICING APPROXIMATION (III)

Application:

the Stein and Stein model with correlation

( )tt Y=σ

( ) ( )∫

−−− =−+=uts

tt deuFemmsM0

2)(,)( θσ αθα

( ) ( )( )dstsFcsMtT

T

t tt ∫ −+−

= 222* )(1ν

( )

( ) ( )∫∫ ∫

∫ ∫

−+

=

−−

−−

T

t

T

t

T

s tsr

t

s

T

t s

T

s

srr

dssFsTFcdsdrrMesMc

FdsYdreYE

)()()( 2

*

α

α

,αλ=c

APPLICATIONS TO OPTION PRICING APPROXIMATION (IV)

Numerical results

)2.0,0953.0,05.0,2.0,4,100ln,5.0( =======− tt rmXtT σλα

APPLICATIONS TO THE STUDY OF LONG-MEMORY VOLATILITY MODELS (I)

Example: long-memory volatilities

( ) ( )

)(~ where

,~10

212

ss

t

st

Yf

dsst

=

−Γ

= ∫−

σ

σβ

σ β

Assume that (see for example Comte, Coutin and Renault (2003)),

( ) ( )

( )( ) drduur

durTdr

T

s

s

u

T

s r

T

sr

∫ ∫

∫ ∫

Γ−+

−+Γ

=

−

0

21

22

~

,~1

1

σβ

σβ

σ

β

β

APPLICATIONS TO THE STUDY OF LONG-MEMORY VOLATILITY MODELS (II)

( ) ( )∫ ∫ −+Γ

=T

s rWs

T

srWs drDrTdrD 2*2* ~

1

1 σβ

σ β

( ) ( )

( )

( ) ( ) ( ) ( ) ( )

−×

+Γ=

−

+Γ=

Λ

∫ ∫

∫ ∫

∫

−−ts

T

t

T

s rrsr

ts

T

t

T

s rWs

t

T

t s

FdsYfdrYfYferTE

FdsdrDrTE

FdsE

'

1

2

~~1

*

2**

*

αβ

β

βραλ

σσβ

ραλ

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY FOR JUMP-

DIFFUSION MODELS WITH STOCHASTIC VOLATILITY (I)

( )

( ) [ ]TtZdBdW

dstkrxX

t

tsss

t

st

,0,1

2

1

0

2

0

2

∈+−++

−−+=

∫

∫

ρρσ

σλ

In Alòs, León and Vives (2007) we considered the following modelfor the log-price of a stock under a risk-neutral probability Q:

independentAdapted to the

filtration generatedby W

( ) ( )∫ ∞<−=

=

dyek

yf

y νλ

λνλ

11

with

,)( measureLévy and

intensity th Poisson wi Compound

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (II)

( )( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )

∂−

−++

Λ∂+

=

∫

∫ ∫

∫

−−

−−

−−

t

T

t ssxtsr

t

T

t R sssstsr

t

t

sssxtsr

tttt

FdsXsBSekE

FdsdyXsBSyXsBSeE

FdsXsGeE

FXtBSEV

νλ

ννν

νρν

,,

,,,,

,,2

,,

0

Hull and White

Correlation

Jumps

Similar arguments as in the previous paper give us thefollowing extension of the Hull and White formula:

( ) ( ) ( ) ( )σσν ,,,,; 2 xtBSxtGtT

Yt xxx

t ∂−∂=−

=

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (III)

After some algebra, we can prove from this expression that:

( ) 0,)( If 22≥−≤

δσ δsrCFDE trs

[ ] ttrttrDD σσ +

↓ =lim

and ( )t

ttt

tt

t kDx

X

I

σλσ

σρ −−→

∂∂ +*

( ) 0,)( If 22 <−≤

δσ δsrCFDE trs

and

( ) ( ) tt

T

t

T

s trs LdrdsFDEtT

σσ δδ

++ →

− ∫ ∫,

2

1( ) ( )

tttt

tt LX

xItT σ

σρ δδ +− −=

∂∂− ,

*

lim

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (IV)

Example 1: classical jump-diffusion models

Assume that the volatility process can be written as

( )( ) ( ) rrrr

rr

dWYrbdrYradY

Yf

,,

,

+==σ

( ) ( )

( ) uusu

r

s

susu

r

srs

dWYDYux

b

YsbduYDYux

aYD

,

,,

∫

∫

∂∂+

+∂∂=

( ) ( )tttt YtbYfD ,'=+σ

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (V)

( ) ( )( )),('1

lim *tt

tt

ttT YtbYfkx

x

I ρλσ

+−=∂∂

→

( ) ( ) ( )r

srr

t

trtr dWecemYmY −−−−

∫+−+= αα α2

If Y is an Ornstein-Uhlenbeck process of the form

( ) ( )( )tt

tt

tT Yfckxx

I'2

1lim * αρλ

σ+−=

∂∂

→

(this agrees with the results in Medvedev andScaillet (2004))

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (VI)

Example 2: fractional stochastic volatility models with H>1/2

Assume that the volatility process can be written as

0=+ttD σ

( ) ( ) ( ) ( )∫

−−−− +−+==r

t

Hr

srtrtrrr dWecemYmYYf αα ασ 2;

( )∫ ∫

−

−−−−r

t s

r

s

Hur dWdusueH 3

1

)(2

1 α

( )t

tt

tT

kx

x

I

σλ−=

∂∂

→*lim

That is, the at-the-money short-datedskew slope is not affected by the

correlation in this case

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (VII)

Example 3: fractional stochastic volatility models with H<1/2

Assume that the volatility process can be written as

( ) ( ) ( ) ( )∫

−−−− +−+==r

t

Hr

srtrtrrr dWecemYmYYf αα ασ 2;

( ) ( )( )

( )s

Hr

t

sr

r

t s

r

s

Hsrur

dWsre

dWdusueeH

2

1

3

1

)(

)(2

1

−−−

−−−−−

−+

−−

−

∫

∫ ∫

α

αα

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (VIII)

( )

.as0

'2

)(

1

2

12

tT

FYfcdrdsD

tT

E t

T

t

T

s trWs

H

→→

−−

∫ ∫−+ασ

( ) ( ) ( )ttt

tHtT Yfcx

X

ItT '2lim *

2

1

αρ−=∂∂

− −→

That is, the introduction of fractional components withHurst index H<1/2 in the definition of the volatilityprocess allows us to reproduce a skew slope of order

( )2

1, −>− δδtTO

More similar to the ones observed in empirical data (see Lee (2004))

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (IX)

Example 4: Time-varying coefficients

(Fouque, Papanicolaou, Sircar and Solna (2004))

Assume that the volatility process can be written as

( )

( ) ( ) ( ) ( )

( ) 0,)(

,2

2

1

>−=

+∫−+=

=

+−

−−−

∫

εα

α

σ

ε

αα

sTs

dWescemYmY

Yf

r

t rsrdss

tr

rr

r

t

Next maturity date

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (X)

( )( )

( )

tT

Yfc

drdsFDEtT

t

T

t

T

s trs

→

+

+−+

−∫ ∫

−+

as zero totends

2

'

2/1

1

2/1

1

1

2

12

εερ

σε

In this case, the short-date skew slope of theimplied volatility is of the order

( ) ε+−− 2

1

tTO

Then

BIBLIOGRAPHY

E. Alòs (2006): A generalization of the Hull and White formula with applicationsto option pricing approximation. Finance and Stochastics 10 (3), 353-365.

E. Alòs, J. A. León and J. Vives (2007): On the short-time behaviour of theimplied volatility for stochastic volatility models with jumps. Finance and Stochastics 11 (4), 571-589

E. Alòs and C. O. Ewald (2008): Malliavin Differentiability of the Heston Volatilityand Applications to Option Pricing. Advances in Applied Probability 40 (1), 144-162.

F. Comte, L. Coutin and E. Renault (2003): Affine fractional stochastic volatilitymodels with application to option pricing. Preprint.

J. P. Fouque, G. Papanicolau, K. R. Sircar and K. Solna (2004): Maturity Cyclesin Implied Volatilities. Finance and Stochastics 8 (4), 451-477.

R. Lee (2004): Implied volatility: statics, dynamics and probabilistic interpretation. Recent advances in applied probability. Springer.

A. Medvedev and O. Scaillet (2004): A simple calibration procedure of stochasticvolatility models with jumps by short term asymptotics. Discussion paper HEC, Gèneve and FAME, Université de Gèneve.