On the impact of correlation on option prices: a Malliavin Calculus - - PowerPoint PPT Presentation

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

• E. Alòs (2006): A generalization of the Hull and White formula with applications

to 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 the

implied 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 and skews observed in real market data:

( )

* 2 * 2

1 2 1

t t t t t

dB dW dt r dX ρ ρ σ σ − + +       − =

= ρ Log-price Volatility (stochastic, adapted to the filtration generated by W) Implied volatility smile Implied volatility skew

≠ 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 the

• ption prices. As an application, we can use it to construct option pricing

approximation formulas, or to study the short-time behaviour of the implied volatility 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 Malliavin derivative and the Skorohod integral to develop adequate change-of-variable formulas for anticipating processes Here our pourpose is to present the basic concepts on Malliavin 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 × = T L h W h W h W f F

n

, in variable random ..., ,

2 2 1

( ) [ ] ( )

{ }

process Gaussian , ,

2

T L h h W ∈

( ) ( ) ( ) ( ) ( ) [ ] ( )

Ω × ∂ ∂ =∑ T L t h h W h W h W x f F D

i n i t

, in Derivative Malliavin ..., ,

2 2 1

(closable operator)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (III) Malliavin derivative: examples

[ ]

) ( 1 Scholes)

• (Black

2

• r

exp

, 2 t

r S S D W t S

t t t W r t

σ σ σ =       +         =

[ ] )

( 1 , t W D

s s W t

=

( )

( ) ( ) [ ]

) ( 1 ) (

,

r ce Y D Uhlenbeck Ornstein dW e c e m Y m Y

t r t t W r r t r t t t − − − − −

= − + − + =

∫

α α α

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (IV) Skorohod integral: definition It is the adjoint of the Malliavin derivative operator:

[ ]

( ) ( ) ( )

h W h T L h

W

= ⇒ ∈ δ ,

2

( )

( ) ( )

S F ds u F D E F u E

s T W s W

∈ =

∫

all for , 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 s W s T s s T s s

ds u F D dW u F dW Fu

The Skorohod integral is an extension of the classical Itô integral

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI) THE ANTICIPATING ITÔ’S FORMULA

( ) ( ) ( ) ( ) ( )( )

, ' ' 2 1 ' '

∫ ∫ ∫

∇ + + + =

t s s s t s s t s s s t

ds u u X F ds v X F dW u X F X F X F

∫ ∫

+ + =

t t s s s t

ds v dW u X X

Non necessarily adapted

( )

dr v D dW u D u u

s r W s s r r W s s s

∫ ∫

+ + = ∇ 2 2 : where

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI)

Proof (sketch)

. assume we simplicity

• f

sake For the ≡ v

( ) ( )

( ) ( )

∑ ∫ ∑ ∫

      +       + =

+ +

2

1 1

' ' 2 1 '

i i i i i i

t t s s t t t s s t t

dW u X F dW u X F X F X F

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

∫

t s ds

u

2

2 1

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VII)

( ) ( ) ( )

∑∫ ∑∫ ∑ ∫

+ + +

− =      

1 1 1

' ' '

i i i i i i i i i

t t s t W s t t s s t t t s s t

ds u X F D dW u X F dW u X F

∫

t s s s

dW u X F ) ( ' 2 1

∫ ∫

     

t s s r r W s s

ds u dW u D X F ) ( ' '

( )

[ ]

∫

− t s s W s

ds u X F D '

AN EXTENSION OF THE HULL AND WHITE FORMULA (I)

[ ]

t t T r t

F H e E V

) ( * − −

=

• ption price

payoff

( )

T T T

X T BS V ν ; , =

The Black-Scholes function final condition implies that

        − =

∫

ds t T v

T t s t 2 2

1 σ

Basic idea Black-Scholes pricing formula Log-price

where

AN EXTENSION OF THE HULL AND WHITE FORMULA (II)

Then

( )

( )

[ ]

t T T t T t T r t

F X T BS E F V E e V ν ; ,

* * ) (

= =

− −

The classical Hull and White term

( )

( )

t t t

F X t BS E ν ; ,

*

is the option price in the uncorrelated case

compare

Then we want to evaluate the difference

( ) ( )

t t T T

X t BS X T BS ν ν ; , ; , −

We need to construct an adequate anticipating Itô’s formula

process ng anticipati an is

t

ν

AN EXTENSION OF THE HULL AND WHITE FORMULA (III)

AN EXTENSION OF THE HULL AND WHITE FORMULA (IV) Anticipating Itô’s formula

( ) ( ) ( ) ( ) ( ) ( )(

)

( )

∫ ∫ ∫ ∫ ∫

∂ ∂ + ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + =

− t s s s t s s s s t t s s s s s s t s s t t

ds u Y X s x F ds u Y D Y X s y x F dY Y X s y F dX Y X s x F ds Y X s s F Y X F Y X t F

2 2 2 2

, , , , , , , , , , , , , ,

∫

=

T t s t

ds Y θ

( )

∫

=

− T s r W s s

dr D Y D θ :

additional term

AN EXTENSION OF THE HULL AND WHITE FORMULA (V)

( )

        − =

− − t t rt t t rt

Y t T X t BS e X t BS e 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 t s s s s s rs t s s s s s rs t s s T t s s rs T t s s s s s BS rs t t rt T T rT T rT

ds s T X s BS e ds Y D s T X s x BS e dZ dW X s x BS e ds X s BS x x L e X t BS e X T BS e V e ν ν σ ν σ σ ν ν σ ρ ρ ρ σ ν ν ν σ ν ν ν

2 2 2 * 2 * 2 2 2 2

, , 2 1 1 , , 2 1 , , , , 2 1 , , , , Black-Scholes differential operator Cancel Zero expectation

AN EXTENSION OF THE HULL AND WHITE FORMULA (VII)

( )

( )

( )

( )

( )

∫

− − − −

− ∂ ∂ ∂ + =

t s s s s s rs t t t rt t T rT

ds Y D s T X s x BS e F X t BS E e F V E e

2 * *

) ( 1 , , 2 , , σ ν ν σ ρ ν

( ) ( )

s s s s

X s H X s x x ν ν , , : , ,

2 2 3 3

=         ∂ ∂ − ∂ ∂

s T s r W s s

dr D σ σ       = Λ

∫

2

*

:

( )

( )

( )

( )

( )

( )

      Λ + = =

∫

− − − − t t s s s t s r t t t t T t T r t

F ds X s H e E F X t BS E F V E e V

* * *

, , 2 , , ν ρ ν

AN EXTENSION OF THE HULL AND WHITE FORMULA (VIII)

( )

( )

      Λ

∫

− − t t s s s t s r

F ds X s H e E

*

, , 2 ν ρ

The above arguments do not requiere the volatility to be Markovian. The main contribution of this formula is to describe the effect of the correlation as the term

APPLICATIONS TO OPTION PRICING APPROXIMATION (I)

( ) ( )

( )

∫ ∫

− =       Λ + =

T t t s t t T t s t t t t aprox

ds F E t T F ds E X t H X t BS V

2 * * * * *

1 , , , 2 , , σ ν ν ρ ν

Consider the approximation

APPLICATIONS TO OPTION PRICING APPROXIMATION (II)

( ) ( )

enough regular and with , f dW dt Y m dY Y f

t t t s s

α λ α σ + − = =

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

( )

α α λ ln 1

2

+ ≤ − C V V

aprox t

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

( )

t t

Y = σ

( )

( )

∫

− − −

= − + =

u t s t t

d e u F e m m s M

2

) ( , ) ( θ σ

αθ α

( )

( )

( )

ds t s F c s M t T

T t t t

∫

− + − =

2 2 2 *

) ( 1 ν

( ) ( )

( )

∫ ∫ ∫ ∫ ∫

− +       =            

− − − − T t T t T s t s r t s T t s T s s r r

ds s F s T F c ds dr r M e s M c F ds Y dr e Y E ) ( ) ( ) (

2 * α α

, α λ = c

APPLICATIONS TO OPTION PRICING APPROXIMATION (IV)

Numerical results

) 2 . , 0953 . , 05 . , 2 . , 4 , 100 ln , 5 . ( = = = = = = = −

t t

r m X t T σ λ α

APPLICATIONS TO THE STUDY OF LONG-MEMORY VOLATILITY MODELS (I) Example: long-memory volatilities

( ) ( )

) ( ~ where , ~ 1

2 1 2 s s t s t

Y f ds s t = − Γ =

∫

−

σ σ β σ

β

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

( ) ( ) ( ) ( )

dr du u r du r T dr

T s s u T s r T s r

∫ ∫ ∫ ∫

        Γ − + − + Γ =

− 2 1 2 2

~ , ~ 1 1 σ β σ β σ

β β

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

( ) ( )

∫ ∫

− + Γ =

T s r W s T s r W s

dr D r T dr D

2 * 2 *

~ 1 1 σ β σ

β

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

            − × + Γ =               − + Γ =       Λ

∫ ∫ ∫ ∫ ∫

− − t s T t T s r r s r t s T t T s r W s t T t s

F ds Y f dr Y f Y f e r T E F ds dr D r T E F ds E ' 1 2 ~ ~ 1

* 2 * * * α β β

β ρ α λ σ σ β ρ α λ

APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY FOR JUMP- DIFFUSION MODELS WITH STOCHASTIC VOLATILITY (I)

( )

( )

[ ]

T t Z dB dW ds t k r x X

t t s s s t s t

, , 1 2 1

2 2

∈ + − + + − − + =

∫ ∫

ρ ρ σ σ λ

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

independent Adapted to the filtration generated by W

( ) (

)

∫

∞ < − = = dy e k y f

y

ν λ λ ν λ 1 1 with , ) ( measure Lévy and intensity th Poisson wi Compound

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

( )

( )

( )

( )

( )

( ) ( ) ( ) ( )

( )

( )

      ∂ −       − + +       Λ ∂ + =

∫ ∫ ∫ ∫

− − − − − − t T t s s x t s r t T t R s s s s t s r t t s s s x t s r t t t t

F ds X s BS e kE F ds dy X s BS y X s BS e E F ds X s G e E F X t BS E V ν λ ν ν ν ν ρ ν , , , , , , , , 2 , ,

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

( ) ( ) (

)

( )

σ σ ν , , , , ;

2

x t BS x t G t T Y t

x xx 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:

( )

, ) ( If

2 2

≥ − ≤       δ σ

δ

s r C F D E

t r s

[ ]

t t r t t r

D D σ σ

+ ↓

= lim and

( )

t t t t t t t

k D x X I σ λ σ σ ρ − − → ∂ ∂

+ *

( )

, ) ( If

2 2

< − ≤       δ σ

δ

s r C F D E

t r s

and

( )

( )

t t T t T s t r s

L drds F D E t T σ σ

δ δ + +

→ −

∫ ∫

, 2

1

( )

( )

t t t t t t

L X x I t T σ σ ρ

δ δ + −

− = ∂ ∂ −

, *

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

( ) ( ) ( )

r r r r r r

dW Y r b dr Y r a dY Y f , , , + = = σ

( ) ( ) ( )

u u s u r s s u s u r s r s

dW Y D Y u x b Y s b du Y D Y u x a Y D , , ,

∫ ∫

∂ ∂ + + ∂ ∂ =

( ) ( )

t t t t

Y t b Y f D , ' =

+σ

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

( )

( ) ( )

) , ( ' 1 lim

* t t t t t t T

Y t b Y f k x x I ρ λ σ + − = ∂ ∂

→

( )

( ) ( )

r s r r t t r t r

dW e c e m Y m Y

− − − −

∫

+ − + =

α α

α 2

If Y is an Ornstein-Uhlenbeck process of the form

( )

( )

( )

t t t t t T

Y f c k x x I ' 2 1 lim

*

α ρ λ σ + − = ∂ ∂

→

(this agrees with the results in Medvedev and Scaillet (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 =

+ t t

D σ

( ) ( )

( ) ( )

∫

− − − −

+ − + = =

r t H r s r t r t r r r

dW e c e m Y m Y Y f

α α

α σ 2 ;

( )

∫ ∫

        −       −

− − − r t s r s H u r

dW du s u e H

3 1

) ( 2 1

α

( )

t t t t T

k x x I σ λ − = ∂ ∂

→ *

lim

That is, the at-the-money short-dated skew 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 H r s r t r t r r r

dW e c e m Y m Y Y f

α α

α σ 2 ;

( ) ( )

( )

( )

s H r t s r r t s r s H s r u r

dW s r e dW du s u e e H

2 1 3 1

) ( ) ( 2 1

− − − − − − − −

− +         − −       −

∫ ∫ ∫

α α α

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

( )

. as ' 2 ) ( 1

2 1 2

t T F Y f c drds D t T E

t T t T s t r W s H

→ →           − −

∫ ∫

− +

α σ

( )

( )

( )

t t t t H t T

Y f c x X I t T ' 2 lim

* 2 1

α ρ − = ∂ ∂ −

− →

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

( )

2 1 , − > − δ

δ

t T O

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

( ) ( )

( )

( )

( )

( )

, ) ( , 2

2 1

> − = + ∫ − + = =

+ − − − −

∫

ε α α σ

ε α α

s T s dW e s c e m Y m Y Y f

r t r s r ds s t r r r

r t

Next maturity date

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

( )

( )

( )

t T Y f c drds F D E t T

t T t T s t r s

→       +       + − + −

∫ ∫

      − +

as zero to tends 2 ' 2 / 1 1 2 / 1 1 1

2 1 2

ε ε ρ σ

ε

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

( )

ε + −

−

2 1

t T O

Then

BIBLIOGRAPHY

• E. Alòs (2006): A generalization of the Hull and White formula with applications

to 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 the

implied 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 Volatility

and Applications to Option Pricing. Advances in Applied Probability 40 (1), 144- 162.

• F. Comte, L. Coutin and E. Renault (2003): Affine fractional stochastic volatility

models with application to option pricing. Preprint.

• J. P. Fouque, G. Papanicolau, K. R. Sircar and K. Solna (2004): Maturity Cycles

in 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 stochastic

volatility models with jumps by short term asymptotics. Discussion paper HEC, Gèneve and FAME, Université de Gèneve.