Stochastic Volatility Jump Diffusion Model for Option Pricing

(1)

Stochastic Volatility Jump-Diffusion Model

for Option Pricing

Nonthiya Makate, Pairote Sattayatham

School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, Thailand E-mail: nonthiyam@hotmail.com, pairote@sut.ac.th

Received July 26, 2011; revised September 2, 2011; accepted September 15, 2011

Abstract

An alternative option pricing model is proposed, in which the asset prices follow the jump-diffusion model with square root stochastic volatility. The stochastic volatility follows the jump-diffusion with square root and mean reverting. We find a formulation for the European-style option in terms of characteristic functions of tail probabilities.

Keywords:Jump-Diffusion Model, Stochastic Volatility, Characteristic Function, Option Pricing

1. Introduction

Let be a probability space with filtration . All processes that we shall consider in this section will be defined in this space. An asset price model with stochastic volatility has been defined by Heston [1] which has the following dynamics:



, , P

 

t 0 t T

 







d t t d td t

S

S S  t v W



, (1)





,

d d v

t t t

v  v t vdW_t

where St is the asset price,  is the rate of return of

the asset, vt is the volatility of asset returns, 0 is

a mean-reverting rate,  is the long term variance,

0

 is the volatility of volatility, and are standard Brownian motions corresponding to the processes t and t, respectively, with constant correlation

S t

W v

t

W

S v

 . In 1996, Bate [2] introduced the jump-diffusion stochastic volatility model by adding log normal jump t

to the Heston stochastic volatility model. In the original formulation of Bate, the model has the following form:

Y





d d S d

t t t t t t t

S S  t v dW S Y N_ S, (2)





d   d  v

t t t

v v t vdW_t ,

where is the Poisson process which corresponds to the underlying asset t, t is the jump size of asset price

return with log normal distribution and t means that there is a jump the value of the process before the jump is used on the left-hand side of the formula. Moreover, in 2003, Eraker Johannes and Polson [3] extended Bate’s

work by incorporating jumps in volatility and their model is given by

S t

N

S Y

S





d d d S

t t t t t t

S S  t v W S Y N d tS, (3)





d d d v v

t t t t v

v  v t v W Z dN_t .

Eraker et al. [3] developed a likelihood-based estima-

tion strategy and provided estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Moreover, they examined the volatility structure of the S&P and Nasdaq indices and indicated that models with jumps in volatility are pre- ferred over those without jumps in volatility. But they did not provide a closed-form formula for the price of a European call option.

In this paper, we would like to consider the problem of finding a closed-form formula for a European call option where the underlying asset and volatility follow the Model (3). This formula will be useful for option pricing rather than an estimation of it as appeared in Eraker’s work.

The rest of the paper is organized as follows. In Sec- tion 2, we briefly discuss the model descriptions for the option pricing. The relationship between stochastic differential equations and partial differential equations for the jump-diffusion process with jump stochastic volatil-ity is presented in Section 3. Finally, a closed-form formula for a European call option in terms of characteristic functions is presented.

2. Model Descriptions

(2)

exists, the asset price t under this measure follows a jump-

diffusion process, and the volatility t follows a pure mean

reverting and square root diffusion process with jump, i.e.

our models are governed by the following dynamics:

S

v









d S d d S

t t d

S

t t t t

S S r m t v_t W S Y N_





, (4)

d d d v d v

t t t t t

v  v t v_t W Z N ,

where St, vt,  , ,  , and are defined as

in Bate’s model, is the risk-free interest rate, and are independent Poisson processes with constant intensities

S t

W W_tv

r S

t

N

v t

N

S

 and v_{respectively. is the jump size}

of the asset price return with density t

Y



Y y



 and

 

_t : m and _t

E Y    Z is the jump size of the volatility

with density _Z . Moreover, we assume that the jump processes and are independent of standard Brownian motions and .



z



t

N

S t

W

X



S t

N v

v t

W

3. Partial Integro-Differential Equations

Consider the process



 1_, 2 where

t  Xt Xt

 



 1

t

X

and  2

t

X are processes in  and satisfy the following

equations:

 



 





   



 

  

1 1 2 1 2

1 1

d t t , t , t , t ,

t t

  

1 _d 1

t

1

d t

d

X f t X X t W

X Y_

 



X X t

N

  

X X t

g

d g

(5)  2



 1 2





 1  2



 

2 2

2

d , , , ,

d

t t t t t

t t

X f t X X t W

Z N

 



2

d _t

where f1,g1, f2 and g2 are all continuously differenti-

able,  1 and are standard Brownian motions

t

W  2

d

t

W

 

with __d 1_,  2_{ }_

 

Corr W_t W_t , N_t 1 and N_t 2 are

independent Poisson processes with constant intensities  1

 and _ 2 _{respectively.}

Since every compound Poisson process can be repre- sented as an integral form of a Poisson random measure [4] then the last term on the right hand side of (5) can be written as follows:

 1  1  1  1

_

_

0 0

d d

t t

s s s n n s Q

 1 1

t

N n

d ,

X Y N_  X Y_  _qJ s q



 

X

 



N n

   

_

_

2 2

1

0 0

d t d

t t

s s n R d ,

Z N Z rJ s

 









 

r

where Yn are i.i.d. random variables with density Y

 

y

and J tQ1 is a Poisson random measure of the process



1

t n

n

with intensity measure ,

N

Q  Y   1 Y

 

d dq t Zn are

i.i.d. random variables with density _Z

 

z , and J_R is a

Poisson random measure of the process with  2

1

t

N t

n

R 





Zn

intensity measure  2

 

_{d d} .

Z r t

 

Let U x x



1, 2



be a bounded real-valued function and

twice continuously differentiable with respect to x1 and 2

x and







 1  2



 1  2

1, ,2 T , T t 1, t 2

u x x t E U X_ X X x X x _(6)

By the two dimensional Dynkin formula [5], is a solution of the partial integro-differential equation (PIDE)

u





_

_



 



 





1 2

(1)

1 2 1 2

(2)

1 2 1 2

, ,

0 , ,

, , , , d

Y

Z

u x x t

u x x t t

u x y x t u x x t y y

u x x z t u x x t z z

 

 



 



 _   _







subject to the final condition u x x T



₁, ,₂



U x x



₁, ₂



.

The notation  is defined by

























1 2 1 2

1 2

2 2

2 1 2 1 2

1 2 1 2

1 2 1

2 2 1 2

2 2

2

, , , ,

, ,

, , , ,

1 2

, , 1

2

u x x t u x x t

Au x x t f f

x x

u x x t u x x t

g g g

x x x

u x x t g

x





 



 

 

  

 



(7)

4. A Closed-Form Formula for the Price of a

European Call Option

Let C denote the price at time t of a European style call

option on the current price of the underlying asset with strike price and expiration time . t

S

K T

The terminal payoff of a European call option on the underlying stock St with strike price K is

max



STK,0



.

This means that the holder will exercise his right only if ST K and then his gain is STK. Otherwise, if



T

S K, then the holder will buy the underlying asset

from the market and the value of the option is zero. Assuming the risk-free interest rate is constant over the lifetime of the option, the price of the European call at time t is equal to the discounted conditional

expected payoff

(3)





 

_

_

 

_

_

_

_

 









( )

, , ; ,

max ,0 ,

, d , d

1

, d

e

( , )d

1 ₍ _{, )d}

[ | , ]

e

t t r T t

T t t

r T t

T T t t T T t t T

K K

t r T t T T t t T

K t r T t

T t t T

K

t T T t t T

T t t K

r T t

C S v t K T

e E S K S v

e S P S S v S K P S S v S

S S P S S v S

S

Ke P S S v S

S S P S S v S

E S S v K

 

 

   

  



 

 

 _  _

 

 _  _

 

 

 _ _

 



 

  

 







 



 

( )

1 2

( , )d

( , , ; , ) ( , , ; , )

T t t T

K

r T t

t t t t t

P S S v S

S P S v t K T Ke P S v t K T



 

 





(8) where E_ is the expectation with respect to the risk- neutral probability measure, P



S S vT t, t



is the cor-

responding conditional density given



S vt, t



and









1 t, , ; ,t T T t, t d T T t, t

K

P S v t K T S P S S v S E S S v



 

 

   _ _





  

Note that 1 is the risk-neutral probability that

(since the integrand is nonnegative and the integral over is one), and finally that

P



 

T

S K



0,









2 , , ; , , d

Pr ob( | , )

t t T t t T

K

T t

P S v t K T P S S v S

S K S v

 

 





t

is the risk-neutral in-the-money probability. Moreover,  

, er T t

T t t t

E__S S v _  S for t0.

Assume that the asset price t and the volatility t

satisfy (4), we would like to compute the price of a European call option with strike price

S v

K and maturity

. To do this, we make a change of variable from t

to , i.e. where satisfies (4) and its inverse

T S

ln 

t

L

t

S S_t

L

t . Denote the logarithm of the strike

price. By the jump-diffusion chain rule, satisfies the SDE

S e klnK

lnSt





ln d d ln 1 d

2



 

_   _   

 

S t S

t t t

v

d S r m t v W Yt NtS(9)

Applying the two-dimensional Dynkin formula [5] for the price dynamics (9) and volatility t in system (4),

we obtain the value of a European-style option, as a

function of the stock log-return denoted by

v

t

L

ln

, ,

t t

L S t vt

















ln

, , ; , ; ,

, , ; , ; , ,

k

t t t

K t

C L v t k T C e v t e T C e v t e T C S t K T

  



i.e.,



_{, , ; ,}



r T t  _max _eLT ,0



,

t t

C l v t k T e  E_ K L l v v_

and satisfies the following PIDE:











 











0 , , ;



, , ; , , , ; , d

S

Y

v

Z

C

C l v t k T

t

C l y v t k T v t k T y y

C l v z t k T v t k T z z

 



 _{ }

 _{  }



 

 _   _

 

 _   _



 

 ,

C l

 

(10) Here the operator  as in (7) is defined by









2 2

2

2 2 , , ; ,

1 2 1 2

S C C

C l v t k T r m v v

l v

C C

v v

l v l

C v

  



 

 

  _  _  

  _ _ _ _

 

 

  

 

 



 





2 1 2

C

v r





 

In the current state variable, the last line of (8) becomes



, , ; ,



l 1



, , ; ,



  2



, , ; ,



k rT t

C l v t k T e P l v t k T e   P l v t k T

(11) where



, , ; ,



:



l, , ; ,k



, 1, 2

j j

P l v t k T P e v t e T j .

The following lemma shows the relationship between and in the option value of (11).

1

P

2

P

Lemma 1 The functions and 2 in the option

value of (11) satisfy the following PIDEs

1

P P















_

_{ }

1 1

1

0 , , ; ,

1 , , : ,

S

S y

Y

P P

P l v t k T v

t l

P

v r m P

v

e P l y v t k T y

 

 



 _{ } 

  _{ } 

 



  



 



_

_   _

 







dy

and subject to the boundary condition at expiration time

tT;





1 , , ; , 1l k

P l v T k T  _ (12) moreover, P2 satisfies the equation





2

2 2

0 P P l v t k T, , ; , rP t

 _{ }

  _{ }  



 

(4)

and subject to the boundary condition at expiration time

;

tT





2 , , ; , 1l k

P l v T k T  _ , (13) The operator  is defined by

 















 











2

2 2

, , ; ,

1 1

:

2 2

1 2

, , ; , , , ; , d

S

Y v

Z

f l v t k T

f f f

r m v v v

l v l

f f

v v rf

l v v



.

f l y v t k T f l v t k T y y

f l v z t k T f l v T k T z z

  

 

 

 

  

 

_   _{ }     

 

 

  

  

 _   _





(14)

Note that 1l k 1 if lk and otherwise 1l k 0.

The following lemma shows how to calculate the functions P1 and P2 as they appeared in Lemma 1.

Lemma 2 The functions and can be calculated by the inverse Fourier transforms of the characteristic function, i.e.

1

P P₂









0

, , ; , 1 1

, , ; , Re d

2 π 



  

    

 

 

 ixk j

j

e f l v t k T

P l v t k T x

ix ,

for with denoting the real component of a complex number. By letting

1, 2

j Re[ ]

T t

  . 1) The characteristic function f1 is given by







 

1 , , ; , exp 1 ixl

f l v t x t  g  vh₁  



,

where

 















1

1 2 2

1 1 1

1 1 1 1

2

1

e h

e

 

 _

  

 

  



    

 





1

S S

g   r m ix m 













 





 





 

1

1 1

2

1

1 2ln 1

2

1 d

ix y S

Y

v

Z

zh

e

e y y

e  z z

 

 _{ }



  

 









  _{ } _  

  

 _ _  _  



 

 _ _ 

 

 



 

,





1 ix 1

   

and  1 122ix ix



1



.

2) The characteristic function f2 is given by







 



2 , , ; ,  exp 2   2    

f l v t x t g vh ixl r ,

where

 

















2

2 2 2

2

2 2 2 2 1

e h

e

 

 _

  



  

 _

     ,

 





2 ix

S

g   r m r 

















 



 



_{ }

2

2 2 2

2 1 2ln 1

2

1 d 1

S ixy v

Y Z

zh

e

e y y e  z

 



dz

  

     

 



  _{ } _  

   

 _ _  _   _



 

 _ _ 

 



_

 

_



2 i x

   

and  ₂ ₂22ix ix



1



.

In summary, we have just proved the following main theorem.

Theorem 3 The value of a European call option of (4) is



, , ; ,



1



, , ; ,



  2



, , ; ,



l k rT t

C l v t k T e P l v t k T e   P l v t k T

where P1 and P2 are given in Lemma 2.

5. Conclusions

This paper has proposed asset price dynamics to accom- modate both jump-diffusion and jump stochastic volatility. Under this proposed model, an analytical solution is derived for a European call option via the characteristic function.

6. Acknowledgements

This research is (partially) supported by The Centre of Excellence in Mathematics, the Commission on Higher Education (CHE).

Address: 272 Rama VI Road, Ratchathewi District, Bangkok, Thailand.

7. References

[1] S. L. Heston, “A Close Form Solution for Options with Stochastic Volatility with Applications to Bond and Cur-rency Options,” The Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343. doi:10.1093/rfs/6.2.327 [2] D. Bates, “Jump and Stochastic Volatility: Exchange Rate

Processes Implicit in Deutsche Mark in Options,” Review of Financial Studies, Vol. 9, No. 1, 1996, pp. 69-107.

(5)

[3] B. Eraker, M. Johannes and N. Polson, “The Impact of Jumps in Volatility and Returns,” The Journal of Finance,

Vol. 58, No. 3, 2003, pp. 1269-1300. doi:10.1111/1540-6261.00566

[4] R. Cont and P. Tankov, “Financial Modeling with Jump

Processes,” CRC Press, Boca Raton, 2004.

(6)

Appendix

Proof of Lemma 1. We plan to substitute (11) into (10).

Firstly, we compute

    1 2 2                 

k r T t k r T t

l P P

C

e e re

t t t P

1   2

1              

 k r T t

l P l P

C

e e P e

l l l

 _  1_   2

  

 



k r T t

l P P

C

e e

v v v

 

2 2

2

1 1

1

2 2 2

      _ _ _ _       

 k r T t

l P l P l

C

e e e P e

l l l 2 2  P l   2 2 2

1 1  

                

k r T t

l P l P

C

e e e

l v l v v l v

2  P   2 2 2 1 2 2 2             

k r T t

l P P

C

e e

v v v2



 



,







 

_

_

 

_

_

1 1 , , ; , , , ; , , , ; , , , ; , l y

k r T t

C l y v t k T C l v t k T

e P l y v t k T

                 





 

_

_





















 

_

_

_













1 2 1 1 1 1 1 1 1 1 , , ; , , , ; , , , ; , , , ; , , , ; , , , ; , , , ; , , , ; ,

1 , , ; ,

, , ; ,       _  _         _ _ _       _   _                 

k r T t l

l y

l l

k r T t

l y

l

e P l v t k T e P l v t k T

e e P l y v t k T P l y v t k T

e P l y v t k T e P l v t k T

e P l y v t k T P l v t k T

e e P l y v t k T

e P



l y v t k T





 

_

_

_

1

2 1

, , ; ,

 k r T t  _  , , ; ,   , , ; ,



_

P l v t k T

e P l y v t k T P l v t k T

and













 

_

_





 

_

_









 

_

_

_

1 2 1 2 1 1 2 2 , , ; , , , ; , , , ; , , , ; , , , ; , , , ; , , , ; , , , ; , , , ; , , , ; ,        



  _    _   _  _    _   _    _   _          

k r T t l

l k r T t

C l v z t k T C l v t k T

e P l v z t k T e P l v z t k T

e P l v t k T e P l v t k T

e P l v z t k T P l v t k T

We substitute all terms above into (10) and separate it by assumed independent terms of and . This gives two PIDEs for the risk-neutralized probability for

: (Equation (15))

1



P P₂

( , , ; , ), 1, 2



j

P l v t k T j

 



 





2

1 1 1 1 1

1 2 1

2 2

2

1 1 1

1 1 1 1

2

1

1 1

0 ( ) 2

2 2

1 ₁ _{, , ; ,} _{, , ; ,} _{( , , ; , )} _d

2 , S S y Y v

P P P P P

r m v P v v P

t l v l l

P P P

v v rP e P l y v t k T P l y v t k T P l v t k T

l v v v

P l v

                      _  __  _   _   _   _ _  _  _     _ _  _  _   _      _        



              

_

_{ }

_{  }

1

, ; , , , ; , Z d

z t k T P l v t k T  z z



  

 







y y (15)

subject to the boundary condition at the expiration time according to (12).



t T

By using the notation in (14), PIDE (15) becomes

















 





1 1 1 1 1 1 1 1

0 , , ; ,

1 , , ; ,

: , , ; ,

S

S y

Y

P P

P l v t k T v

t l

P

v r m P

v

e P l y v t k T y P

P l v t k T t      _{ }    _{ }            _    _{ }   _{ } 



           dy  

For P l v t k T₂



, , ; ,



:





2 2 2 1 0 2 S P P

rP r m v v

t  l  

     _   _         







 







 

2 2 2

2

2 2 2

2 2 2 2 2 2 2 1 1 2 2

, , ; , , , ; , d

S

Y

v

Z

P P P

v v v rP

l v

l v

P l y v t k T P l v t k T y y

P l v z t k T P l v t k T z z

                      _   _    _   _



       



(16) subject to the boundary condition at the expiration time



t T according to (13). Again, by using the notation in

(14), PIDE (16) becomes

2

2 2

0  [ ]( , , ; , ) 



 

P

P l v t k T rP t  2 P v    2 2 2 :  [ ]( , , ;    P

P l v t k T

(7)



The proof of Lemma 1 is now completed. For j1, 2 the characteristic functions for



, , ; ,





j

P l v t k T ,

with respect to the variable k are defined by



, , ; ,



: ixkd



, , ; ,

j j

f l v t x T e P l v t k T

 

 

_

 _,

with a minus sign to account for the negativity of the measure dPj.

Note that fj also satisfies similar PIDEs



, , ; ,



0

j

j j

f

f l v t x T t



 

 _{ } 

  , (17)

with the respective boundary conditions













, , ; , d , , ; ,

d

ixk

j j

ixk ixl



f l v t x T e P l v t k T

e k l

e



   

 

   







k

since dP l v T k Tj



, , ; ,



d1l k dH l



k



 



kl



dk.

Proof of Lemma 2

1) To solve for the characteristic function explicitly, letting  T t be the time-to-go, we conjecture that

the function f₁ is given by







 



1 , , ; ,  exp 1   1  

f l v t x t g vh ixl (18)

and the boundary condition

 

1 0 0 1 0

g  h .

This conjecture exploits the linearity of the coefficient in PIDE (17).

Note that the characteristic function f1 always exists.

In order to substitute (18) into (17), firstly we compute

 





1

1 1 1

f

g vh f

t  

 _ _

  

 1 1

  

f ixf l

 

1 1

f h

v 

 _

 f1 2

2 1

1 2

f

x f l

 _{ } 

 

2 1

1 f

ixh f

l v  1

 _

 

 

2 2 1

1 1 2

f

h f

v 

 _















1

1 , , ; , , , ; ,

1 , , ; ,

ixy

f l y v t x t f l v t x t e f l v t x t



 



  

  











 



1







1 1

1

, , ; , , , ; ,

1 , , ; ,

zh

f l v z t x t f l v t x t

e  f l v t x t

 



   

  

and













     









1 1

1

1 , , ; , 1

1 , , ; ,

y y

y ixy

g vh ixl y

e f l y v t x t e e

e e f l v t x t

 



  

    

  

Substituting all the above terms into (17) and after canceling the common factor of f₁, we get a simplified

form as follows:

 









 





 



1



 

1 1

2 1

2 2

1 1

1

1 0

2 1 2 1

2

1 d

1 d .

S

S S

Y v

Z

y

zh ix

g vh r m v ix

v v h vx

vixh vh m

e y y

e  z z

  

   

    

 

 



 

 

   _   _

 

   

  

 



By separating the order and ordering the remaining terms, we can reduce it to two ordinary differential equa-tion (ODEs),

v

 

2 2

 





 

2

1 1 1

1 1

1

2 2

h    h    ix  h   ix1

2x (19)

and

 





 





 



1



 

1 1

1

1 d

1 d .

S S

S

Y v

Z

y

zh ix

g h r m ix

e y

e  z

    

 

   



    

 



m y

z

(20)

Let ₁



ix 1



 and substitute it into (19). We get

 













2 2 1

1 1 2 1 2

2 2 1 1

1 2

2 2 1 1

1 2

2 1 1 1 1

1 2 1 2

2

1 1 ₁

2

2 4 4 1

1 2

2 2

2 4 4 1

2 1

2

h h h ix ix

ix ix h

h h

  

     



   

 



 

 

  _    _

 

 _ _ _ 

 

 

 

 

 _ _ _ 

 

 

 

 

   

  

 _  _  _

  

where 2 2





1 1  ix ix 1

    .

By the method of variable separation, we have

2 1

1 1 1 1

1 2 1 2

2d _{ }_d

 

 

      _  _ 

  

  

h

h h

(8)

Using partial fractions, we get

1

1 1 1 1

1

1 2 1 2

1 1 1

d d

 

 

 

 

 

 _{ } _{ }

  _  _

 

h

h h



 .









0

, , ; , 1 1

, , ; , Re d

2 π

j j

ixk

e f l v t k T

P l v t k T x

ix 

 _  _

   

 

 



 ₍₂₁₎

for j1, 2.

To verify (21), firstly we note that

 

   

_

_













ln ln _ln _,

, d ,

d , , ; , d

, , ; , .

t

t t t

t t j

j

ix St K

ix ix L k

ixk ixl ixk ixk

ixk

l k

E e S L v v

E e L l v v e P l v t k T

e e P l v t k T e e l k k

e f l v t k T



 

 

 



 



 _ _ 

 

 

 _   _

  









Integrating both sides, we obtain

1 1 1 2

1 1 1 1 2 ln

h

C h



 _

 

 

 _ 

 

  

 _{ } 

  

 

. , ; ,



Using boundary conditionh₁(0) 0 , we get 1 1

1 1 ln

C 



   

  _{ } 

 .

Then Solving for h1, we obtain

 















1

1 2 2

1 1

1 ₂

1 1 1 1

1

e h

e

 

 

  



   

     .

In order to solve g1

 

 explicitly, we substitute h1

 



into (20) and integrate with respect to  on both sides. Then we get

 

















 





 





 

1

1 1

2

1

1 2 ln 1

2

1 d

S S

S

Y

v

Z

y

zh ix

g r m ix m

e

e y y

e  z z



   



 _{ }



  



 







  

      

   

    

 

 _  _ 

 

 



Proof of 2). The details of the proof are similar to case 1). Hence, we have





 













0

ln ln

, , ; ,

1 1 _Re _d

2 π

ln ,

1 1 _Re _d

2 π

1 1 _Re _d _,

2 π

sin

1 1 _d _,

2 π

1 _sgn 1

2 π

j

t t t

t t

ixk

ix S_t K

ixl k

e f l v t k T

x ix

E e S L v v

x ix

e

E x L l v

ix

x l k

E x L l v

x

E l k







 

  

 

 







 



2 , , ; , exp 2 2

f l v t x t  g  vh  ixlr

v

  _ _ 

 

 

  _ _

 

 

   

 

     

 _ _ 

 

  

     

 

 

  









0

sin _d _,

1 1_sgn _,

2 2

1 ,

t t

l k t t

x

x L l v v

x

E l k L l v v

E L l v v







 

 

 

 

 

 _     _

 

 

 _   _







2

where g2

   

 ,h2  , and 2 are as given in the Lemma.

We can thus evaluate the characteristic functions in explicit form. However, we are interested in the risk-neutral probabilities . These can be inverted from the characteristic functions by performing the following integration

, 1, 2

j

P j 

where we have used the Dirichlet formula

0

sinx_d ₁

x x  





and the function is defined as if , 0 if

sgn 0

 

sgn x 1 x0