short-time asymptotic methods in financial mathematics · short-time asymptotics of option prices...

Short-Time Asymptotic Methods in Financial

Mathematics

José E. Figueroa-López

Department of Mathematics

Washington University in St. Louis

Probability and Mathematical Finance Seminar

Department of Mathematical Sciences

Carnegie Mellon University

February 20, 2017

J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 1 / 39

Outline

1 Short-time asymptotics of Option Prices

Short-time ATM Skew Asymptotics

A Calibration Case-Study

Ongoing and Future Research

2 High-Frequency Based Estimation Methods

Multipower Variations and Truncated Realized Variations

Optimal Threshold Selectionvia Expected number of jump misclassifications

via conditional Mean Square Error (cMSE)



Short-time asymptotics of Option Prices

Outline











Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics

Outline











Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics


Theorem (F-L & Ólafsson, F&S 2016)

For Y ∈ (1,2), the implied volatility σ̂ (κ, t) of a vanilla option with

log-moneyness κ and time-to-maturity t is such that

∂σ̂ (κ, t)∂κ

∣∣∣∣κ=0∼

d1t−

12 , if pure jump model with C+ 6= C−,

d1t12−

1Y , if pure jump model with C+ = C−,

d1t1−Y

2 , if mixed model with C+ 6= C−,

d1, if mixed model with C+ = C−.

Furthermore, in the asymmetric cases, the sign of d1 is the same as

that of C+ − C−.


Short-time asymptotics of Option Prices A Calibration Case-Study

Outline












Calibration of S&P 500 Option Prices

Objectives:

• To assess the plausibility of the divergence of the skew to −∞ as

a power of time-to-maturity.

• Calibrate the parameter Y , which measures the activity of small

jumps in the price process.

Data:

• Daily closing bid and ask prices for S&P 500 index options, across

all strikes K , and maturities t from Jan. 2 to Jun. 30, 2014.



Implied Volatility Smile and ATM Skew

Time-to-maturity (t)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ATM

skew

-12

-10

-8

-6

-4

-2

(a)

log-moneyness (κ)-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

Implied

vol.

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14(b)

Figure: (a) The graphs of the ATM implied volatility skew are consistent with

divergence to −∞ as a power law. (b) The implied volatility smiles on Jan.

15, 2014, corresponding to maturities ranging from 0.032 (8 days) to 0.25 (3

months), show greater steepness as time-to-maturity decreases, which is

consistent with the presence of jumps.J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 8 / 39


Calibration of the Index Of Jump Activity Y

DayJan Feb Mar Apr May Jun

Ypj

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2(a)

1-day regression

21-day regression


Ycc

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2(b)

1-day regression

21-day regression

Figure: The Y -estimates implied by daily and monthly regressions of

log ∂σ̂(κ,t)∂κ

∣∣∣κ=0

on log t assuming a pure-jump model (panel (a)), and a mixed

model (panel (b)). The first and third quantiles of the regressions’ R2 are

0.977 and 0.995, respectively.



ATM Implied Volatility

Time-to-maturity (t)0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

ATM

implied

vol.

0.06

0.08

0.1

0.12

0.14

0.16

0.18(a)


Volatility

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22(b)

VIX

ATM vol.

Figure: (a) ATM implied volatility as a function of time-to-maturity in years, for

each business day in Jan–Jun, 2014. (b) ATM implied volatility for the

shortest outstanding maturity compared to the VIX index. These suggest a

mixed model rather than a pure jump model since σ̂(t ,0)→ σ0 > 0.


Short-time asymptotics of Option Prices Ongoing and Future Research

Outline











Short-time asymptotics of Option Prices Ongoing and Future Research


1 Besides the skew, the convexity of the implied volatility,∂2σ̂(κ,t)∂κ2

∣∣∣κ=0

, is very important in trading and, thus, its asymptotic

behavior is of great interest.

2 Small-time asymptotics of options written on Leveraged

Exchange-Traded Fund (LETF). LETF is a managed portfolio,

which seeks to multiply the instantaneous returns of a reference

Exchange-Traded Fund (ETF).

3 There is no known results for American options. It is expected that

the early exercise feature won’t affect the leading order terms of

the asymptotics but they should contribute to the high-order terms.


High-Frequency Based Estimation Methods

Outline











High-Frequency Based Estimation Methods

Merton Log-Normal Model

Consider the following model for the log-return process Xt = log StS0

of a

financial asset:

Xt = at + σWt +∑

i:τi≤t

ζi , ζii.i.d.∼ N(µjmp, σ

2jmp), {τi}i≥1 ∼ Poisson(λ)

Goal: Estimate the volatility σ based on a discrete record Xt1 , . . . ,Xtn

0.00 0.02 0.04 0.06 0.08

1.0

1.1

1.2

1.3

Log-Normal Merton Model

Time in years (252 days)

Sto

ck P

rice

Price processContinuous processTimes of jumps


High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations

Outline












Two main classes of estimators

Precursor. Realized Quadratic Variation:

QV [X ]n :=1T

n−1∑i=0

(Xti+1 − Xti

)2

1 Bipower Realized Variations (Barndorff-Nielsen and Shephard):

BPV [X ]n :=n−1∑i=0

∣∣Xti+1 − Xti

∣∣ ∣∣Xti+2 − Xti+1

∣∣ ,2 Truncated Realized Variations (Mancini):

TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).





QV [X ]n :=1T

n−1∑i=0

(Xti+1 − Xti

)2 n→∞−→ σ2 +∑

j:τj≤T

ζ2j


BPV [X ]n :=n−1∑i=0

∣∣Xti+1 − Xti

∣∣ ∣∣Xti+2 − Xti+1


TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).





QV [X ]n :=1T

n−1∑i=0

(Xti+1 − Xti

)2 n→∞−→ σ2 +∑

j:τj≤T

ζ2j


BPV [X ]n :=n−1∑i=0

∣∣Xti+1 − Xti

∣∣ ∣∣Xti+2 − Xti+1

∣∣ ,

2 Truncated Realized Variations (Mancini):

TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).





QV [X ]n :=1T

n−1∑i=0

(Xti+1 − Xti

)2 n→∞−→ σ2 +∑

j:τj≤T

ζ2j


BPV [X ]n :=n−1∑i=0

∣∣Xti+1 − Xti

∣∣ ∣∣Xti+2 − Xti+1


TRVn[X ](ε) :=n−1∑i=0

(Xti+1 − Xti

)2 1{∣∣∣Xti+1−Xti

∣∣∣≤ε}, (ε ∈ [0,∞)).



Calibration or Tuning of the Estimator

Problem: How do you choose the threshold parameter ε?

0.00 0.02 0.04 0.06 0.08

0.85

0.90

0.95

1.00



Sto

ck P

rice


0.00 0.01 0.02 0.03 0.04

0.0

0.1

0.2

0.3

0.4

0.5

Performace of Truncated Realized Variations

Truncation level (epsilon)

Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)

TRVTrue Volatility (0.4)Realized Variation (0.51)Bipower Variation (0.42)

Figure: (left) 5-min Merton observations with σ = 0.4, σjmp = 3(h), µjmp = 0,

λ = 200; (right) TRV estimates for all the truncation levels.





0.00 0.02 0.04 0.06 0.08

0.85

0.90

0.95

1.00



Sto

ck P

rice


0.00 0.01 0.02 0.03 0.040.0

0.1

0.2

0.3

0.4

0.5



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)

TRVTrue Volatility (0.4)Realized Variation (0.51)Bipower Variation (0.42)

Figure: (left) 5-min Merton observations with σ = 0.4, σjmp = 3(h), µjmp = 0,

λ = 200; (right) TRV estimates for all the truncation levels.J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 17 / 39




0.00 0.02 0.04 0.06 0.08

0.800.850.900.951.00



Sto

ck P

rice


0.000 0.005 0.010 0.015 0.020 0.025 0.0300.0

0.1

0.2

0.3

0.4



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)

TRVTrue Volatility (0.2)Realized Variation (0.4)Bipower Variation (0.26)Cont. Realized Variation (0.19)

Figure: (left) 5 minute Merton observations with σ = 0.2, σjmp = 1.5(h),

µjmp = 0, λ = 1000; (right) TRV performance wrt the truncation levelJ.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 18 / 39


Popular truncations ε

Literature consists of mostly “ad hoc" selection methods for ε, aimed to

satisfy sufficient conditions for the consistency and asymptotic

normality of the associated estimators.

The most popular is the so-called Power Threshold:

εPwrα,ω := α hω, for α > 0 and ω ∈ (0,1/2).

The rule of thumb says to use a value of ω close to 1/2 (typically,

0.495).


High-Frequency Based Estimation Methods Optimal Threshold Selection

Outline












General Approach

1 Fix a suitable and sensible metric for estimation error; say,

MSE(ε) = E

[(1T

TRVn(ε)− σ2)2]

2 Show the existence of the optimal threshold ε?n minimizing the

error function;

3 Characterize the asymptotic behavior ε?n (when n→∞) to infer

• qualitative information such as rate of convergence on n and

dependence on the underlying parameters of the model (σ, σJ , λ)• Devise a plug-in type implementation of ε∗ by estimating those

parameters (if possible).



Via Expected number of jump misclassifications

1 Estimation Error: (F-L & Nisen, SPA 2013)

Lossn(ε) := En∑

i=1

(1[|∆n

i X |>ε,∆ni N=0] + 1[|∆n

i X |≤ε,∆ni N 6=0]

).

2 Notation:

Nt := # of jumps by time t

∆ni X := Xti − Xti−1

∆ni N := Nti − Nti−1 = # of jumps during (ti−1, ti ]



Existence and Infill Asymptotic Characterization

Theorem (F-L & Nisen, SPA 2013))

1 For n large enough, the loss function Lossn(ε) is convex and,

moreover, enjoys a unique global minimum ε?n.

2 As n→∞, the optimal threshold sequence (ε?n)n is such that

ε∗n =

√3σ2hn log

(1hn

)+ h.o.t.,

where hn := ti − ti−1 is the time step.

Remark: The leading order term is proportional to the Lévy’s modulus

of continuity of the Brownian motion!.



A Feasible Implementation based on ε?n

(i) Get a “rough" estimate of σ2 via, e.g., realized QV:

σ̂2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2

(ii) Use σ̂2n,0 to estimate the optimal threshold

ε̂?n,0 :=(

3 σ̂2n,0hn log(1/hn)

)1/2

(iii) Refine σ̂2n,0 using thresholding,

σ̂2n,1 =

1T

n∑i=1

|Xti − Xti−1 |21[|Xti−Xti−1 |≤ε̂

?n,0

](iv) Iterate Steps (ii) and (iii):

σ̂2n,0 → ε̂?n,0 → σ̂2

n,1 → ε̂?n,1 → σ̂2n,2 → · · · → σ̂2

n,∞





σ̂2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂?n,0 :=(


)1/2


σ̂2n,1 =

1T

n∑i=1


?n,0

]

(iv) Iterate Steps (ii) and (iii):

σ̂2n,0 → ε̂?n,0 → σ̂2

n,1 → ε̂?n,1 → σ̂2n,2 → · · · → σ̂2

n,∞





σ̂2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂?n,0 :=(


)1/2


σ̂2n,1 =

1T

n∑i=1


?n,0


σ̂2n,0 → ε̂?n,0 → σ̂2

n,1 → ε̂?n,1 → σ̂2n,2 → · · · → σ̂2

n,∞



Illustration I

0.00 0.02 0.04 0.06 0.08

0.85

0.90

0.95

1.00

1.05



Sto

ck P

rice

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

0.0

0.1

0.2

0.3

0.4



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)


Figure: (left) Merton Model with σ = 0.4, σjmp = 3(h), µjmp = 0, λ = 200;

(right) TRV performance wrt the truncation level. Red dot is σ̂n,1 = 0.409,

while purple dot is the limiting estimator σ̂n,∞ = 0.405



Illustration II

0.00 0.02 0.04 0.06 0.08

0.96

1.00

1.04



Sto

ck P

rice

0.00 0.01 0.02 0.03 0.04

0.00

0.10

0.20

0.30



Tru

nca

ted

Re

aliz

ed

Va

ria

tio

n (

TR

V)


Figure: (left) Merton Model with σ = 0.2, σjmp = 1.5(h), µjmp = 0, λ = 1000;

(right) TRV performance wrt the truncation level. Red dot is σ̂n,1 = 0.336,

while purple dot is the limiting estimator σ̂n,∞ = 0.215



Via conditional Mean Square Error (cMSE)

1 We now propose a second approach in which we aim to control

MSEc(ε) := E

(TRVn(ε)−∫ T

0σ2

s ds

)2∣∣∣∣∣∣σ, J

.This is in the more general semimartingale setting:

Xt = X0 +

∫ t

0σsdWs + Jt

where J is a pure-jump process.

2 Key Assumptions:

σt > 0, ∀ t , and σ and J are independent of W .



Key Relationships

bi (ε) := E[(∆iX )21{|∆i X |≤ε}|σ, J

]

Then, with the notation

mi := ∆ni J = Jti − Jti−1 , σ2

i :=

∫ ti

ti−1

σ2s ds,

we have

bi (ε) = − σi√2π

(e− (ε−mi )2

2σ2i (ε+ mi ) + e

− (ε+mi )2

2σ2i (ε−mi )

)+ (m2

i + σ2i )

∫ ε+miσi

ε−miσi

e−x2/2√

2πdx

dbi (ε)

dε=

ε2ai (ε), with ai (ε) :=e− (ε−mi )2

2σ2i + e

− (ε+mi )2

2σ2i

σi√

2π



Key Relationships

bi (ε) := E[(∆iX )21{|∆i X |≤ε}|σ, J

]Then, with the notation


i :=

∫ ti

ti−1

σ2s ds,

we have

bi (ε) = − σi√2π

(e− (ε−mi )2

2σ2i (ε+ mi ) + e

− (ε+mi )2

2σ2i (ε−mi )

)+ (m2

i + σ2i )

∫ ε+miσi

ε−miσi

e−x2/2√

2πdx

dbi (ε)

dε=

ε2ai (ε), with ai (ε) :=e− (ε−mi )2

2σ2i + e

− (ε+mi )2

2σ2i

σi√

2π



Key Relationships

bi (ε) := E[(∆iX )21{|∆i X |≤ε}|σ, J

]Then, with the notation


i :=

∫ ti

ti−1

σ2s ds,

we have

bi (ε) = − σi√2π

(e− (ε−mi )2

2σ2i (ε+ mi ) + e

− (ε+mi )2

2σ2i (ε−mi )

)+ (m2

i + σ2i )

∫ ε+miσi

ε−miσi

e−x2/2√

2πdx

dbi (ε)

dε= ε2ai (ε), with ai (ε) :=

e− (ε−mi )2

2σ2i + e

− (ε+mi )2

2σ2i

σi√

2π



Conditional Mean Square Error (MSEc)

Theorem (F-L & Mancini, 2017)

1 MSEc(ε) := E[

(TRVn(ε)− IV )2∣∣∣σ, J] is such that

d MSEc(ε)

dε= ε2G(ε),

where G(ε) :=∑

i ai(ε)(ε2 + 2

∑j 6=i bj(ε)− 2

∑j σ

2j

).

2 Furthermore, there exists an optimal threshold ε??n that minimizes

MSEc(ε) and is such that G(ε??n ) = 0.



Asymptotics: FA process with constant variance

Theorem (F-L & Mancini, 2017)

1 Suppose that σt ≡ σ is constant and J is a finite jump activity

process (with or without drift; not necessarily Lévy):

Xt = σWt +

Nt∑j=1

ζj

2 Then, as n→∞, the optimal threshold ε??n that minimizes the

cMSE is such that

ε??n ∼

√2σ2hn log

(1hn

), hn → 0.



A Feasible Implementation based on ε??n

(i) Get a “rough" estimate of σ2 via, e.g., the QV:

σ̃2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2

(ii) Use σ̃2n,0 to estimate the optimal threshold

ε̂??n,0 :=(

2 σ̃2n,0hn log(1/hn)

)1/2

(iii) Refine σ̃2n,0 using thresholding,

σ̃2n,1 =

1T

n∑i=1


??n,0


σ̃2n,0 → ε̂??n,0 → σ̃2

n,1 → ε̂??n,1 → σ̃2n,2 → · · · → σ̃2

n,∞





σ̃2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂??n,0 :=(


)1/2


σ̃2n,1 =

1T

n∑i=1


??n,0

]

(iv) Iterate Steps (ii) and (iii):

σ̃2n,0 → ε̂??n,0 → σ̃2

n,1 → ε̂??n,1 → σ̃2n,2 → · · · → σ̃2

n,∞





σ̃2n,0 :=

1T

n∑i=1

|Xti − Xti−1 |2


ε̂??n,0 :=(


)1/2


σ̃2n,1 =

1T

n∑i=1


??n,0


σ̃2n,0 → ε̂??n,0 → σ̃2

n,1 → ε̂??n,1 → σ̃2n,2 → · · · → σ̃2

n,∞



Illustration II. Continued...

0.00 0.02 0.04 0.06 0.08

0.96

1.00

1.04



Sto

ck P

rice

0.00 0.01 0.02 0.03 0.04

0.00

0.10

0.20

0.30



Tru

nca

ted

Re

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ed

Va

ria

tio

n (

TR

V)


Figure: (left) Merton Model with λ = 1000. Red dot is σ̂n,1 = 0.336, while

purple dot is the limiting σ̂n,k = 0.215. Orange square is σ̃n,1 = 0.225, while

brown square is the limiting estimator σ̃n,∞ = 0.199



Monte Carlo Simulations

Estimator σ̂ std(σ̂) Loss ε N

RQV 0.3921 0.0279

σ̂n,1 0.29618 0.02148 70.1 0.01523 1

σ̂n,∞ 0.23 0.0108 49.8 0.00892 5.86

σ̃n,1 0.265 0.0163 62.6 0.0124 1

σ̃n,∞ 0.211 0.00588 39.1 0.00671 5.10

BPV 0.2664 0.0129

Table: Estimation of the volatility σ = 0.2 for a log-normal Merton model

based on 10000 simulations of 5-minute observations over a 1 month time

horizon. The jump parameters are λ = 1000, σJmp = 1.5√

h and µJmp = 0.


High-Frequency Based Estimation Methods Ongoing and Future Research

Outline













1 In principle, we can apply the proposed method for varying

volatility t → σt by localization; i.e., applying it to periods where σ

is approximately constant.

2 However, is it possible to analyze the asymptotic behavior of

MSEc = ε2G(ε) in terms of certain estimable summary

measures? Say,

G(ε) ∼ G0(ε,m1, . . . ,mn, σ, σ̄),

where σ := inft≤T σt and σ̄ := supt≤T σt




3 As it turns, for a Lévy process J and constant σ, the expected

MSE(ε) := E[(TRVn(ε)− IV )2

]is such that

ddε

MSE(ε) = nε2E[a1(ε)](ε2 + 2(n − 1)E[b1(ε)]− 2nhnσ

2)

Therefore, there exists a unique minimum point ε̄??n which is a

solution of the equation

ε2 + 2(n − 1)E[b1(ε)]− 2nhnσ2 = 0.




3 Furthermore,

• As expected, in the finite jump activity case, it turns out

ε̄??n ∼

√2σ2hn log

(1hn

)• But, surprisingly, if J is a Y -stable Lévy process (IA),

ε̄??n ∼

√(2− Y )σ2hn log

(1hn

)Thus, the higher the jump activity is, the lower the optimal threshold

has to be to discard the higher noise represented by the jumps.• Does this phenomenon holds for the minimizer ε??n of the cMSE?• Can we generalize it to Lévy processes with stable like jumps?




3 Furthermore,


ε̄??n ∼

√2σ2hn log

(1hn

)

• But, surprisingly, if J is a Y -stable Lévy process (IA),

ε̄??n ∼


(1hn






3 Furthermore,


ε̄??n ∼

√2σ2hn log

(1hn


ε̄??n ∼


(1hn


has to be to discard the higher noise represented by the jumps.

• Does this phenomenon holds for the minimizer ε??n of the cMSE?• Can we generalize it to Lévy processes with stable like jumps?




3 Furthermore,


ε̄??n ∼

√2σ2hn log

(1hn


ε̄??n ∼


(1hn


has to be to discard the higher noise represented by the jumps.• Does this phenomenon holds for the minimizer ε??n of the cMSE?

• Can we generalize it to Lévy processes with stable like jumps?




3 Furthermore,


ε̄??n ∼

√2σ2hn log

(1hn


ε̄??n ∼


(1hn





Further Reading I

J.E. Figueroa-López, R. Gong, and C. Houdré,

High-order short-time expansions for ATM option prices of exponential

Lévy models.

Mathematical Finance, 26(3), 516-557, 2016.

J.E. Figueroa-López and S. Ólafsson,

Short-time expansions for close-to-the-money options under a Lévy jump

model with stochastic volatility.

Finance & Stochastics 20(1), 219-265, 2016.

J.E. Figueroa-López and S. Ólafsson,

Short-time asymptotics for the implied volatility skew under a stochastic

volatility model with Lévy jumps.

Finance & Stochastics 20(4), 973-1020, 2016.



Further Reading II

J.E. Figueroa-López & J. Nisen.

Optimally Thresholded Realized Power Variations for Lévy Jump

Diffusion Models.

Stochastic Processes and their Applications 123(7), 2648-2677, 2013.

J.E. Figueroa-López & C. Mancini.

Optimum thresholding using mean and conditional mean square error.

Available at https://pages.wustl.edu/figueroa, 2017.

J.E. Figueroa-López, C. Li, & J. Nisen.

Optimal iterative threshold-kernel estimation of jump diffusion processes.

In preparation, 2017.


short-time asymptotic methods in financial mathematics · short-time asymptotics of option prices...

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