STOCHASTIC VOLATILITY MODELS

(1)

On the impact of correlation on option prices: a

Malliavin Calculus approach

(2)

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

(3)

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

r dt dW dB

dX σ  + σ ρ + − ρ



 



 −

=

= 0 ρ

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

Implied volatility smile Implied volatility skew

ρ ≠ 0

(4)

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 option 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.

(5)

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

(6)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (II)

Malliavin derivative : definition

( ) ( ) ( )

( )

[ ]

( ^× ^Ω )

=

T L

h W h

W h

W f

F

_n

, 0 in

variable random

..., ,

2 2 1

( ) ( [ ] )

{ }

process Gaussian

, 0 , h L

²

T h

W ∈

( ) ( ) ( )

( ) ( )

[ ]

( ^× ^Ω )

∂

= ∑ ∂

T L

t h h W h

W h

x W F f

D

_n _i

i t

, 0 in

Derivative Malliavin

..., ,

2 2 1

(closable operator)

(7)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (III)

Malliavin derivative: examples

[ ]

( ) 1

Scholes) -

(Black - 2

r exp

, 0 2

t

r S

S D

W t

S

t t t

W r

t

σ σ σ

=

 



 

  +





 

= 

[ ]

( ) 1

₀_,

t W

D

_t^W _s

=

_s

( ) ^{( )}

( ) 1 [ ] ( ) ) (

, 0 0 0

r ce

Y D

Uhlenbeck Ornstein

dW e

c e

m Y

t r

t t

W r

r

t t r

t t

−

=

−

+

− +

= ∫

α

α α

(8)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (IV)

Skorohod integral: definition

It is the adjoint of the Malliavin derivative operator:

[ ]

( ^T ) ( ) ^h ^W ( ) ^h

L

h ∈

²

⁰ ^, ⇒ δ

^W

=

( ( ) û ^F ) Ê ( ^D ^F ) û ^ds ^F ^S

E ^δ

^W

= ∫

₀^T _s^W _s

^, ^for ^all ∈

Example:

(9)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (V)

Skorohod integral: properties

The Skorohod integral of a process multiplied by a random variable

( )

∫

₀^T

^Fu

^s

^dW

^s

⁼ ^F

₀^T

^u

^s

^dW

^s

⁺

₀^T

^D

^W^s

^F ^u

^s

^ds

The Skorohod integral is an extension of

the classical Itô integral

(10)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI)

THE ANTICIPATING ITÔ’S FORMULA

( ) ( ) ( )

( ) ^' ^' ( )( ) ^,

2 ' 1

'

0 0

∫

∇ +

+

=

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 _s _s ^t _s

t

X u dW v ds

X

0 0 0

Non necessarily adapted

( ) û û ^D û ^dW

^s

^D ^v

r

^dr

W

s s

r r

W s s

s

⁼ ⁺ ∫ ⁺ ∫

∇ : 2

0

2

0

where

(11)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI)

Proof (sketch)

. 0 assume

we simplicity of

sake

For the v ≡

( ) ( ) ( )

( )

∑ ∫

 

 

 + 

 

 

 + 

=

+

2 0

1

2 '' 1

'

i

i i

i

i i

t

t s s

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

∫

₀^t

^u

^s²

^ds

2

1

(12)

SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VII)

( )

( ) _∑∫ ( )

∑∫

∑ ∫

+ +

+

−

=

 

 





1 1

1

' '

'

i

i i

i

i i

i

i i

t

t t s

W s t

t t s s

t

t s s

t

ds u X

F D

dW u

X F

dW u

X F

∫

₀^t

^F ^' ⁽ ^X

^s

⁾ ^u

^s

^dW

^s

2 1

∫

₀^t

^F ^'' ⁽ ^X

^s

⁾ ^ _ ^ ∫

₀^s

^D

^s^W

^u

^r

^dW

^r

^ _ ^ ^u

^s

^ds [ ( ) ]

∫

₀^t

^D

^s^W⁻

^F ^' ^X

^s

^u

^s

^ds

(13)

AN EXTENSION OF THE HULL AND WHITE FORMULA (I)

[

^r ^T ^t ^t

]

t

E e H F

V =

^* ⁻ ⁽ ⁻ ⁾

option price payoff

(

^T ^T

)

T

BS T X

V = ^, ^; ν

The Black-Scholes function final condition implies that

 





 





= _T − _t ∫ ^ds

v

T

t s t

2

1 σ

Basic idea

Black-Scholes

pricing formula Log-price

where

(14)

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

( )

( ^BS ^t ^X

^t ^t

^F

^t

)

E

^*

^, ^; ν

is the option price in the uncorrelated case

co m p ar e

(15)

Then we want to evaluate the difference

( ^T ^X

^T ^T

) ^BS ( ^t ^X

^t ^t

)

BS ^, ^; ν − ^, ^; ν

We need to construct an adequate anticipating Itô’s formula

process ng

anticipati an

t

is ν

AN EXTENSION OF THE HULL AND WHITE FORMULA (III)

(16)

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

t

s s t

t

ds u Y X x s

F

ds u Y D Y

X y s

x F

dY Y

X y s

dX F Y

X x s

F

ds Y

X s s

Y F X F

Y X t F

0

2 2

2 0

2

0 0

0 0 0

, ,

0 ,

,

∫

=

^T

t s

t

ds

Y θ ( ) ^D

⁻

^Y

^s

^: ⁼ _∫

_s^T

^D

^s^W

^θ

^r

^dr

additional

term

(17)

AN EXTENSION OF THE HULL AND WHITE FORMULA (V)

( ) _ ^





 





=

⁻

−

t t

rt t

t

rt

Y

t X T

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

(18)

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 BS s s s s s

rs

t t rt

T T rT

T rT

s ds X T

BS s e

ds Y

s D X T

x s e BS

dZ dW

X x s

e BS

ds X

s x BS

L x e

X t BS e

X T BS e

V e

ν

ν ν σ

σ

ν σ σ ν

ρ

ρ ρ

σ ν

ν ν

σ ν

ν ν

2 2

0

2

* 2

* 2 2 2

2

, 2 ,

1 , 1 2 ,

1 ,

,

, 2 ,

1 , , ,

,

Black-Scholes differential operator Cancel

Zero expectation

(19)

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

s D X T

x s e BS

F X

t BS E

e F

V E e

0

2

*

) (

, 1 2 ,

, , ν σ σ ν

ρ

ν

(

s Xs s

)

H

(

s Xs s

)

x

x³₃ ²₂  ^, ^,

ν

=^: ^, ^,

ν









∂

− ∂

∂

s T

s r

W s

s

= ^ _ ^ D σ dr ^ _ ^ 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

0

*

, 2 ,

, ,

ρ ν

ν

(20)

AN EXTENSION OF THE HULL AND WHITE FORMULA (VIII)

( )

( ) ^ _ ^

 

 ∫

^t

^e

⁻^r ^s⁻^t

^H ^s ^X

^s ^s

Λ

^s

^ds ^F

^t

E

0

*

, ,

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

(21)

APPLICATIONS TO OPTION PRICING APPROXIMATION (I)

( )

∫ ( )

∫

= −

 

 



 Λ

+

=

T

t s t

t

t T

t s

t t

t t aprox

ds F t E

T

F ds E

X t H

X t BS V

2

*

1 , ,

2 ,

, ,

σ ν

ρ ν

ν

Consider the approximation

(22)

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 1 ln

2

+

≤

− V C

V

_t _aprox

A similar result was proven by Alòs and Ewald

(2008) for the Heston volatility model

(23)

APPLICATIONS TO OPTION PRICING APPROXIMATION (III)

Application:

the Stein and Stein model with correlation

( σ

t

= Y

t

)

( ⁻ )

⁻ ^{( )}⁻

⁼ ∫

⁻

+

=

_t ^s ^t ^u

t

s m m e F u e d

M

0

)

2

( , )

( σ

^α ^αθ

θ

( ) ( ^M ^s ^c ^F ^{( )} ^s ^t ) ^ds

t T

T

t t

t^* ²

⁼ ¹ ₋ ∫

²

⁽ ⁾ ⁺

²

⁻

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

(24)

APPLICATIONS TO OPTION PRICING APPROXIMATION (IV)

Numerical results

) 2 . 0 ,

0953 .

0 ,

05 . 0 ,

2 . 0 ,

4 ,

100 ln

, 5 . 0

( T − t = X

_t

= α = m = λ = r = σ

_t

=

(25)

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

Example: long-memory volatilities

( ) ( )

)

~ ( where

~ , 1

0

1 2 2

s s

t

s t

Y f

ds s

t

= Γ −

= ∫

⁻

σ β σ

σ

^β

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

( ) ( )

( )

( ) ^u ^du ^dr

r

du r

T dr

T s

s

u T

s r

T r s

∫ ∫

 





 





Γ + −

+ −

= Γ

− 0

2 1

2 2

~

~ , 1

1 β σ β σ σ

β

(26)

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

( ) ( )

∫

_s^T

^D

^s^W^* ^r²

^dr ⁼ _Γ ₊ ∫

_s^T

^T ⁻ ^r ^D

^s^W^*

^~

^r²

^dr

1 1 σ

σ β

^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

*

β α

β

ρ α λ

σ β σ

ρ

α

λ

(27)

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

( )

( ^dW ^dB ) ^Z ^t ^{[ ]} ^T

ds t

k r

x X

t

t s

s s

t s t

, 0 ,

1 2 1

0

2 0

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

(28)

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

( )

( ( ) ( ) ) ( )

( )

( ) ^ _ ^

 

 ∂

−

 

 



 + −

+

 

 



 ∂ Λ

+

=

∫

∫ ∫

∫

−

t T

t x s s

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

ρ ν

ν

, ,

, 2 ,

, ,

0

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 ^; ^G ^t ^, ^x ^,

²

^BS ^t ^, ^x ^, t

T

t Y

^t

= ∂

_xx

− ∂

_x

= −

(29)

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

²

  ≤ −

²

≥

  ^D σ ^F ^C ^r ^s

^δ

δ E

_s _r _t

[ ]

^t ^r ^t ^t

t

r_↓

D σ = D

⁺

σ

lim

and ( )

t t

t t t

t

k

D X x

I

σ σ λ

σ

ρ ₋

−

∂ →

∂

* ₊

( ) ⁽ ⁾ ^, ⁰

If

²

  ≤ −

²

<

  ^D σ ^F ^C ^r ^s

^δ

δ E

_s _r _t

and

( )

_t^T _s^T

^E ( ^D

^s ^r

^F

^t

) ^drds ^L

^t ^t

t

T

₊_δ

σ →

^δ ⁺

σ

− ¹

²

∫ ∫

^,

( ) ( )

t t t t

t

L

X x t I

T σ

σ ρ

_δ

δ +

−

= −

∂

− ∂

^* ^,

lim

(30)

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

)

s u u r

s

s u

s u r

r s s

dW Y

D Y x u

b

Y s b du Y D Y x u

Y a D

,

, ,

∫

∂ + ∂

∂ +

= ∂

( ) ( )

t t t

t

f Y b t Y

D

⁺

σ = ' ,

(31)

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

( ) ¹ ⁽ ^' ^{( )} ⁽ ^, ⁾ ⁾

lim

^* _t _t

t t

T

x k f Y b t Y

x

I λ ρ

σ ⁺

−

∂ =

∂

→

( )

^{( )} ^{( )} r

s r r

t t

r t

r

m Y m e c e dW

Y ⁼ ⁺ ⁻

⁻^α ⁻

⁺ ∫ ² ^α

⁻^α ⁻

If Y is an Ornstein-Uhlenbeck process of the form

( ) ( ^{( )}

^t

)

t t

T

x k c f Y

x

I 1 2 '

lim

^*

λ ρ α

σ ⁺

−

∂ =

∂

→

(this agrees with the results in Medvedev and

Scaillet (2004))

(32)

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

+ t

D

t

σ

( ) ⁼ ⁺ ( ⁻ )

⁻ ^{( )}⁻

⁺ ∫

⁻ ^{( )}⁻

=

^r

t

H r s

r t

r r

r

f Y Y m Y m e

^α

c α e

^α

dW

σ ^; ²

( )

∫ ∫  





 



 −

 

 



 −

^r ⁻ ⁻ ⁻

t s

r s

u H

r

u s du dW

e

H

³

1

) 2 (

1

_α

( )

t t

T

x k x I

σ

− λ

∂ =

∂

→ *

lim That is, the at-the-money short-dated

skew slope is not affected by the

correlation in this case

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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 r

r

f Y Y m Y m e

^α

c α e

^α

dW

σ ^; ²

( ) ( )

( )

s H

r t

s r

r

t s

r s

s H r u

r

dW s

r e

dW du

s u e

e H

2 1

3 1

) (

) 2 (

1

− −

−

− −

−

− +

 





 



 − −

 

 



 −

∫

∫ ∫

α

α α

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

( )

. as

0 ' 2

) (

1

2 2 1

t T

F Y

f c

drds D

t T

E

^T _t

t T

s r t

W H s

→

 







 







−

− ∫ ∫

−

+

σ α

( ) ( )

^t

^{( )}

^t

t H t

t

T

x c f Y

X t I

T 2 '

lim

² ^*

1

= − ρ α

∂

−

⁻

∂

→

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

− ^t

^δ

δ T

O

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

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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

>

−

=

∫ +

− +

=

+

−

− −

∫

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 ds r

s t

r

r r

r t

Next maturity date

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

( ) ( )

( )

t T

Y c f

drds F

D E t

T

t T

s s r t

→

 

 





 +



 





+ + −

− ∫ ∫



 



 − +

as zero to

tends

2 ' 2

/ 1

1 2

/ 1

1 1

2 2 1

ε ρ ε

ε

σ

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

( ^T ⁻ ^t )

⁻²¹⁺^ε

O

Then

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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.

STOCHASTIC VOLATILITY MODELS

On the impact of correlation on option prices: a

Malliavin Calculus approach

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