Recent Developments in Option Pricing

Loading....

Loading....

Loading....

Loading....

Loading....

Full text

(1)

doi:10.4236/jmf.2011.13009 Published Online November 2011 (http://www.SciRP.org/journal/jmf)

Recent Developments in Option Pricing

Hui Gong1, Aerambamoorthy Thavaneswaran2, You Liang2 1

Department of Mathematics & Computer Science, Valparaiso University, Valparaiso, USA 2

Department of Statistics, University of Manitoba, Winnipeg, Canada E-mail: hughgong@gmail.com, thava@temple.edu, umlian33@cc.umanitoba.ca Received September 1, 2011; revised September 28, 2011; accepted October 10,2011

Abstract

In this paper, we investigate recent developments in option pricing based on Black-Scholes processes, pure jump processes, jump diffusion process, and stochastic volatility processes. Results on Black-Scholes model with GARCH volatility (Gong, Thavaneswaran and Singh [1]) and Black-Scholes model with stochastic volatility (Gong, Thavaneswaran and Singh [2]) are studied. Also, recent results on option pricing for jump diffusion processes, partial differential equation (PDE) method together with FFT (fast Fourier transform) approximations of Pillay and O’ Hara [3] and a recently proposed method based on moments of truncated lognormal distribution (Thavaneswaran and Singh [4]) are also discussed in some detail.

Keywords:Stochastic Volatility, Black-Scholes Partial Differential Equations, Option Pricing, Monte Carlo

1. Introduction

Option pricing is one of the major areas in modern fi- nancial theory and practice. Since Black, Scholes, and Merton introduced their path-breaking work on option pricing, there has been explosive growth in derivatives trading activities in the worldwide financial markets. The main contribution of the seminal work of Black and Scholes [5,6] and Merton [7] was the introduction of a preference-free option pricing formula that does not in- volve an investor’s risk preferences and subjective views. Due to its compact form and computational simplicity, the Black-Scholes formula enjoys great popularity in the finance industries and is based on the strong assumption that the volatility of the stock returns is constant. How- ever, implied volatility of the stock prices suggests sto- chastic volatility models are more appropriate to model the stock price. The most popular approach is to use the Heston model (Heston [8,9]), which assumes that the underlying asset follows the Black-Scholes model but the volatility is stochastic and follows the Cox Ingersoll Ross process (Cox, Ingersoll and Ross [10]). The empirical results of Bakshi, Cao and Chen [11] suggest that taking stochastic volatility into account is important in option pricing. Motivated by the theoretical considerations, Scott [12], Hull and White [13,14], Ritchken and Trevor [15], and Wiggins [16] generalized the Black-Scholes model to allow stochastic volatility.

Heston and Nandi [17] first provided a solid theoretical

foundation based on the concept of locally risk-neutral valuation relationship for option valuation under nonlin-ear GARCH models using characteristic functions. Heston and Nandi [17], Elliot, Siu and Chan [18], Christoffersen, Heston and Jacobs [19], and Mercuri [20] among others derived closed form option pricing formula under various GARCH models. Recently, Badescu and Kulpeger [21], Barone-Adesi et al. [22,23], and Gong, Thavaneswaran and Singh [1] including others have studied option pric-ing under GARCH volatility. Thavaneswaran, Peiris and Singh [24] and Thavaneswaran, Singh and Appadoo [25] studied option pricing using the moment properties of the truncated lognormal distribution. Gong, Thavaneswaran and Singh [1,2] studied Black-Scholes models with GARCH volatility and with stochastic volatility as in Taylor [26]. They carried out extensive empirical analysis of the European call option valuation for S & P 100 index and showed that the proposed method outperformed other compelling stochastic volatility pricing models. In Tha- vaneswaran and Singh [4], option pricing for jump diffu- sion process with stochastic volatility was studied by view- ing option pricing as a truncated moment of a lognormal distribution. Pillay and O’ Hara [3] studied the FFT based option pricing under a mean reverting process with sto- chastic volatility and jumps by using the PDE approach.

(2)

H. GONG ET AL. 64

,

2, we study option pricing for pure jump processes, jump diffusion models, stochastic volatility models, and jump diffusion models with stochastic volatility. Moreover, closed form solutions are obtained by solving the two-dimensional partial differential equations for stock price in some ex- amples. Section 3 closes the paper with conclusions.

2. Option Pricing and Partial Differential

Equations

Consider the stock and bond model as in Steele [27]

 

 

 

 

dX t  t X t, dt( ,t X t )dW t and

 

 

 

d tr t X t,  t d , t where all of the model coefficients

 

t X t,

 , , and are given by

explicit functions of the current time and current stock price. First, we will use the coefficient matching method to show that arbitrage price at time t of a European op-tion with terminal time T and payout

 

t X t,

r t X t

,

 

 

,

h t X t is given by f t X t

,

 

where f t x

 

, is the solution of the partial differential equation (PDE)

     

   

   

2

1

, , , , ,

2

, ,

t x xx

f t x r t x xf t x t x f t x r t x f t x

 

(2.1)

with terminal condition

,

  

.

f T xh x

If we let denote the number of units of stock that we hold in the replicating portfolio at time t and let denote the corresponding number of units of the bond, then the total value of the portfolio at time t is

 

a t

 

b t

     

( )

 

.

V ta t X tb tt

The condition where the portfolio replicates the con-tingent claim at time T is simply

 

 

V Th X T

d . From the self-financing condition

 

       

dV ta t dX tb t d t (2.2) and the models for the stock and bond, we have

 

 

 

 

 

 

 

 

 

d , ,

, d .

 

 

V t a t t X t b t r t X t t t

a t t X t W t

(2.3)

Then from our assumption that

 

h t X t

,

 

 

,

V t for

any twice differentiable function f t x and the Itô for-mula, we have (2.4).

When we equate the drift and diffusion coefficients from Equation (2.3) and Equation (2.4), we find a simple expression for the size of the stock portion of our repli-cating portfolio:

 

t x

t,

, af X t

and find

 

 

 

 

 

 

 

 

 

 

2

, , ,

1

, , ,

2

, , .

x

t xx

x

t X t f t X t b t r t X t t

f t X t f t X t t X t

f t X t t X t

 

 

The 

t X t,

 

fx

t X t,

 

terms cancel, and b t

 

is given by

 

 

, 1

 

t

,

 

21 xx

,

 

2

,

 

b t

f t X t f t X t t X t

r t X tt

 

 .

Because V t

 

is equal to both f t X t

,

 

and

       

a t X tb tt , the values for a t

 

and b t

 

give us a PDE for f t X t

,

 

:

 

 

 

 

 

 

2

 

1

, ,

, 1

, , ,

2

x

t xx

f t X t f t X t X t

r t X t

f t X t f t X tt X t

 

 

 .

Now, when we replace X t

 

by x, we arrive at the general Black-Scholes PDE (2.1) and its terminal bound-ary condition (2.2).

We can solve the Black-Scholes PDE to get the time-t call price. The link between the stochastic solution to PDE and the martingale approcach is given by the fol- lowing Feynman-Kac Theorem.

Theorem 2.1 A functionf t x

,

defined by

 

  d

 

 

, ,

T tV X

f t x  e  h X T X tx

 

where dX t

 

t X t,

 

dt

t X t,

 

dW t

 

satis-fies partial differential equation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

2

1

d , d , , d , d

2

1

, , , , , d , , d

2

t xx x

t xx x x

V t f t X t t f t X t t X t t f t X t X t

. f t X t f t X t t X t f t X t t X t t f t X t t X t W t

  

  

 

 

 

(3)

 

   

   

   

2

1

, , , , ,

2

, ,

t x xx

f t x t x f t x t x f t x

V t x f t x

 

 

with the terminal condition f T x

,

  

h x .

oles PDE Consider the Black-Sch

 

,

 

, 1 2 2

 

,

 

2

t x xx , ,

f t xrxf t x   x f t xrf t x

with the terminal condition: (a) f T x

,

 

xK

: 0,

h  

; (b) ; (c)

T x,

  

h x for some function

,

f

f T xx; (d) f T x

,

log

 

x .

holes equation has a stochasti

The above the Black-Sc c

solution of the form

 

 

 

 

, e r T t ,

f t xX t  x

where, under eutral risk measu f X T  

  

the n re 

 

 

 

 

d  d  d ,

,

X t rX t t X t W t

and X t

 

is a geometric Brownian motion of the form

 

 

2

2

exp .

X TX t r T tW Tt

For specific terminal conditions, the closed form time-

se (a les formula

t price can be obtained by finding the stochastic solutions. In Ca ), the time- t price is given by the

Black-Scho- 

 

 

 

   

 

 

 

2

2

log 2

( )

1 2

,

e e d

2

d e d ,

x r T t T t z z

r T t

f t x

2

e 1 e

r T t Q

r T t

X T K X t x

K z

x K

 

     

  

 

  

   

where

  

  

 

 

2

1 2

log

d x K r 2 T t , d T t.

T t

 

  

 d1

In Case (b), the time-t price can be represented as

 

 

 

   

 

 

 

22

e e

e e .

T t W T t

r T t Q

x

22

r T t T t Z

r T t

,

f t

f X X x

f x

t      t

 

   

 

 

 

 

In Case (c), for some real v,

 

   

    

 

 

 

   

 

 

   

 

 

2

2

2

1 2

(

,

e e

e .

r T t T t W T t

Q

W T t

r T t

f t x

22

e r T t e r T t

X t X t x

x

x

    



  

     

   



 

In Case (d),

 

   

  

 

 

 

 

 

   

 

 

 

22

2

,

e log e

e log 2 .

T t W T t r T t

Q r T t f t x

X t X t

t x

x T

  

 

     

 

   

 

 

   

2.1. Option Pricing for Pure Jump Processes (PDE Approach)

In order to price European options based on jump processes we need to know the evolution of both (Vecer [28]),

 

Y

X t and YX

 

t X

 he ev jump

in order to determine both martin- asures and . It is possible to preserve the ry of t olution of the prices with the

preserves the direction: when gale me

symmet tion that the

Y

 

Y

X t jumps up, YX

 

t jumps down, and vise versa. The jump

 

N t ualized

ated w

bel e pair and Y; it cannot be indi

t n rast to the geom

it Y. case of Poiss

on o

h t gs to th one asset i

he asset of cont

In the

vi- etric Brow-

olution, d

nian motion model, where the noise factor WX is asso- ciated with the asset X, and the noise factor Y

W is asso-

ci on ev

it is the intensity  of the Poisson process that is asso- ciated with the particular asset. Under the X measure,

 

N t has intensity Xt and the process N t

 

Xt is a martingale, w under Y

measure,

hile the N t

 

has in-

tensity Yt and the process N t

 

Yt is a martingale. The price process XY

 

t driven by a SDE of the form

 

e 1

 

 

Y

,

Y Y

dX   X td N  t has a solution as a geometric Poisson process

 

 

 

tt

0 exp  e 1 .

  

Y Y

X t X N t (2.5)

The inverse price process YX

 

t satisfy the SDE of

the form

Yt

 

 

 

dYX t  e  YX t d N t e t , and has a geometric Poisson p ess representation

 roc

1 X

 

 

0

 

e 1

X

.

X X

Y tY exp N t     t (2.6) Y

 and X

The values of are linked by the rela-tionship XeY

, where  is the size of the jump of

 

logXY t . Let V be a contract that pays off

 

Y Y

f X T

the con units of an asset Y at time e price of with respect to the reference Y is given by

T. Th tract V

 

Y Y

 

.

VY t t f

XY T 

(4)

H. GONG ET AL. 66

we can write

 

,

 

Y

 

.

u t x  f X T X tx

Similarly we can also com ute the price of this con- respect to a reference asset X as

 

X

X

 

,

X t X

V t  f Y T

Y Y Y

t Y

p tract with

where fX

is a payoff function in terms of the asset X. And the price function is defined as

 

 

 

, X u t x

, ( .

X X Y

t X X

u t x  f Y T Y t x

Then the following theorem (Vecer [28]) g e form of the PDEs fo

)

ives th

r and

Theorem 2.2 ice sed on e geo-

metric Poisson pr

 

, X u t x (a) The pr

 

, Y u t x .

function ba th ocess (2.5),

 

,

 

( )

X X X

t X X

u t x  f Y T Y tx

satisfies 

 

,

X t u t x

u

 

 

with the terminal condition

nction ba the geom ric Poisson pro- ce

, e , e 1 , 0

X X X X

t t x u t x xux t x

 

 

    

,

 

.

X X

u T xf x (b) The price fu sed on et

ss (2.6), uY

 

t x, YtfY

XY

 

T

XY

 

tx satisfies

 

,

, e

,

Y Y Y Y

ut t x  ut txu t x

e 1

xuYx

 

t x, 

 

0

erminal condition

Proof of this theorem is given in [28] (pp. 249-250). For a geometric Poisson model (Vecer [28]) for two

where the price follows

, with the t

,

 

.

Y Y

u T xf x

no-arbitrage assets X and Y,

 

 

d  e 1  d Y

Y Y

X t X t N t t

an

in terms of a asset Y is

d by letting N t

 

 m k, the price of a contract that pays off VTI

N

 

Tk

T reference

n by

Y give

 

 

 

 

 

 

 

k m

! .

YT t k m Y

e   Tt

Y Y

Y Y

t t

V t V T N T k

N T k N T N t k m

 

  

    

 

Moreover, the price of using the reference asset X is given by

YtY  t

 

X

V t

 

  

 

e  

.

YT t Y k m

X t Y X

T t

V V t Y

 

!

X

Y

k

t t

m

2.2. Option Pricing for Jump Diffusion Processes Recently Pillay and O’ Hara [3] have studied the FFT sed option pricing under a mean reverting process with stochastic volatility and jumps by finding a closed form of expression for a conditional characteristic function of the log asset process and then apply the FFT method to

e.

ba-

compute the option pric Let

,,

row enerated b a risk neutral

be a probability space on which are de- fined two B nian motion processes. Let be the fil-

tration g y these Brownian mo se th

is probability under which the asset t

tions. Suppo at 

price process S t

 

and volatility process v t

 

are gov- rned by the following dynamics:

e

 

 

 

 

 

 

 

   

 

 

 

 

1

2

1 2

d ln d d ,

d d d ,

d d d ,

 

  

  

S k S S t v W

v b a

t t t t t t

t t t t t

t

v

W t

v t W

W t

where 

 

t is a deterministic function that represents the equilibrium mean level of the asset against time, k is the mean reverting intensity of the asset, a t

 

is a determi- nistic function that describes the equilibrium mean level of the volatility process against time and b is the mean reversion speed of the volatility process. The constant  is the volatility coefficient of the volatility process, and

 

1

W t and W t2

 

are correlated with correlation

coef-ficient  .

On the probability space

,,

, a Poisson process

 

N t in

is further defined for with a constant tensity parameter

all 0 t T ,

0

 . The process N t

 

is assumed pende

to be inde nt of both W t1

 

and W t2

 

. Furthermore,

a sequence of random variables eJi for 1 i N t

 

is

defined to represent the jump sizes of the Poisson proc-ess. Each of the eJi’s are log-normally, identically and

independently distributed over time, where ~

 

, 2 i

J N  

and  0. Then by defining the following process

 

ln

 

X tS t and applying the Itô-Doebin formula to the two-factor mean reverting process with stochastic volatility and jumps we have

 

 

 

 

 

 

1

d 2

eJ 1 ,

t

t v t t

X k t v W

k k

X t t

 

 

   

 

 

(2.7)

 

 

 

d d

d t m

X

N

 

2

dv tb(av t )dt v t dW t , (2.8)

 

 

1 2

(5)

 

 

E e

t u

  iuX .

t T

The method of Wong and Lo [29] has applied to com- pute the characteristic function of the process (2.7).

Duffie, Pan and Singleton [30] introduced a generalized Feynman-Kac theorem for affine jump diffusion proc- esses. By defining the following function:

; , ,

exp

 

,

 

,

 

, iux

, f ut xvB t TC t T xD tT v where

 

, C t T

 

,

 

 

 

 

 

; , , E e | ,

e E e e | , ,

 

 

 

 

iuX

iuX

r rT

T

T T

t t

t t

f u t x v X x v v

 

 

X x v v

(2.10) contingent claim that pays at time T, where r is a constant interest rate, which can be viewed as a

rT iux

e

 

X t is the mean reverting asset price process with jumps defined by (2.7) and v t

 

is the volatility process specified by (2.8), the generalized Feynman-Kac theorem implies that f u t x v

; , ,

solves the following partial in- tegro-differential equation (PIDE):

 

2

1

2 2

; , , ; , , d 0,

t x xx

1 2

v

xv

v m

vv

f k x f vf b a v f v

k k

vf f u t x J v f u t x v q J J

 

  

 

      

 

 

  

variab

f

(2.11) where q J

 

is the distribution function of the random le J and 0 is the constant intensity parameter Poisson process N t

 

.

The coefficients, of the

k v 2km kx and v t

 

, of the mean reverting asset price process (2.7) and the coefficients, b a

v t

 

and  v

 

t of the volatility

process (2.8) are all affine in nature. It follows that the function f u t x v

; , ,

solution of (

is of exponential affine form, and hence the 2.11) has the form

,

B t T and D t

,T

are deterministic

functions of t. From (2.10), it is clear that

; , ,T x v

exp iux ,

 

which is the terminal condition of PIDE ( f u

2.11). This im plies that

-

,T

0,

T T,

0,

,T

0.

B TCD T  (2.12)

Solving the PIDE (2.11) with the terminal conditions (2.12), the conditional characteristic function of the mean reverting process (2.7) with stochastic volatility (2.8) is

 

exp

B t T

 

, C t T x

 

, D t T v

 

, iux

t u

   

, , , and

,

B t T C t T D t T

where

are given i

2.3. Option Pricing for Stochastic Volatility Models twice-differentiable continu

n (2.13). The detail proof of the results is given in Pillay and O’ Hara [3].

Given a ous function

f

; , ,t x v

t T , the price process X t

 

and the volatility rocess v t

 

follow the following stochastic volatility P

processes

 

 

 

1 dX t  x dt x dW t

X

,

 

 

 

2

 

dv t  v dt v dW t ,

ln

 

,

t S t

 

 

dW t dW t dt

where 1 2  . Then the PDE of f can

be obtained by using Itô formula, see (2.14). Setting the drift term to zero we have

 

x

 

   

 

 

0.

2 2

1 1

2 2

v xv

xx

f x f x f x v f

x f

    

 

  

  v fvv  (2.15)

 

 

 

 

  

 

 

   

 

 

 

 

*

2 * * * *

*

2

2 1

* * 1

*

, e 1 , d , , e 1 ,

1 , , 1, 1,

2 2

, ,

1

, , (1) 2

,

k T t

T k T t

t

b e

k m

B t T iu ab D s T s T t C t T iu

k

u a

i a b a b

k b

e k

U

a b V

U k

a

V

k T t

( ) ,

k T t b T t

k T t Ve

D t T Ue

d

u

  

 

  

 

 

 

 

 

    

 

  

      

   

 

 

 

 

 

 

     

 

 

 

2 2 2

*

* 2

1

1 1 * *

2

2 2

* * 1

, 1

2 4

, , 1 , , e 1 .

1 1

, ,

u

iu u k

b b

b k

k

e a b

k u

a b

u

  

  

 

  

 

   

 

 

 

 

(2.13)

 

 

 

 

2

 

2

 

 

 

 

1 2

1 1

d ( ) d d

2 2

        

      

  

x v xv xx vv xv

(6)

H. GONG ET AL. 68

One can calculate the asset price by inverting its con- ditional characteristic function . The con- ditional characteristic function

 

i |

t

S T

 

of the X t

 

process is de- fined as

 

E ei X T  .

t u t

   

Furthermore, if one defining the following function

; , ,

E ei X T  ( ) ,

 

,

ft x vX tx v tv

then solution of (2.15) is the characteristic function. To solve for the characteristic function explicitly, consider an exponential affine form

; , ,

exp

 

 

 

ft x vB  CxDv i x  (2.16)

where  T t and B

0

C

0

D

0

0. Take derivatives of (2.16) with respect to x, v and

, we get

 

x v

f C i f t x v

f

  

 

  

 

  

2

2

; , , ; , ,

; , , ; , ,

vv

D f t x v

f C i f t x v

f D f t x v

 

 

 

; , ,

; , , .

xv

f C i D f t x v

xx

f   B  CxDv ft x v

  

 

  

  

 

   

 

 

(2.17)

Substituting (2.17) into (2.15), it yields

 

 

 

 

   

     

 

   

2

   

2 2 2

( ) ( )( )

1 1

0.

2 2

B C D v x C i

x C i v D

  

( )

x

v D x v C i D

  

      

 

   .18)

  

    

  

   

(2

Equation (2.18) leads to a system of ODEs. We can get the characteristic function by solving this ODE system.

As an example, for the Stochastic Volatility studied by Christoffersen, Heston and Jacobs [1 fined by

model 9]

de- 

 

 

 

 

 

 

 

 

ln

 

,

1

2

d ( )d d ,

d ( )d d ,

X t R uv t t v t W t v t a cv t tv t W t

  

  

where d

X tS t

 

 

1 2

dW t dW t  t,

 

 

 

 

 

 

 

 

 

 

   

 

 

 

2 2 2

0

1 1

2 2

,

(

)

v t u C i cD C i D

C i D D C x

R C i aD B

 

t

      

     

   

    

    

   

and the system of ODEs becomes

 

 

 

 

 

2 2 2

 

 

0

1 1

,

2 2

u C i cD C i D

C i D D

      

    

    

    (2.19)

 

0 C  , (2.20)

 

 

 

0R C  i aD  B  . (2.21) It is clear from (2.20) and C

0

0 that C

 

 0. Hence, (2.19) and (2.21) turn out to be

 

 

 

 

 

 

2 2 2

1 1

,

2 2

.

D ui cD i D D

B Ri aD

        

  

    

 

One can solve these two ODEs under the conditions

 

0 0

B  and D

 

0 0 to obtain

 

d 

and we know the equations

that x  R uv t , 

 

xv t

 

, 

 

v  a cv t

 

and 

 

v  v t

 

. Substitute them into (2.16), we have

2

1

d 2 ln ,

1

a ge

Ri i c

g

    

  

       

 

 

 

2 d d

d 1

, 1

B

i c e

D

ge

  

 

  

 

where

2 2

2

d

, d 2 .

d i c

g i c

i c ui

     

   

   

  

FFT method can be

the conditional characteristic function.

Option Pricin

with Stochastic Volatility

Thavaneswaran and Singh [4] considered the price proc-ess

The used to obtain the call price

from

2.4. g for Jump Diffusion Model

 

S t following a conditional jump diffusion model

 

 

 

   

d log ( ) d ( )d ( ) d ( ),

log S t log S

y t ,

1 1

S t v t t W t Y N t t

t Z t

S t S t

 

 

  

 

where

 

  

 

2

2

v rm t , meY 1, and W t

 

is a standard Brownian mo is a sta process, and the number of j

tion, 

 

t umps N t

 

on

tionary

(7)

 

   

 

 

 

 

 

1

log ,

log log ( ) ,

1 1

N t i i S t

vt t W t Y

S

S t S t

y t t Z t

S t S t

             

where ’s are identically distributed independent nor- m

i Y

al random variables having mean  and variance 2

.

 

 

Let  t 2 t 2n2 T. Then conditioning on 

 

t g the e n as in Theo

and tak xpectatio rem 1 in [4

call price is given by

in ], the

 

 

 

1 1 , 1

e e ( )

!

n

n

m T r T

t n

C S T

m T

S T K

n              

 

 

 

1 1 1 ! n m T t n m T e S n           

 

 

 

1 1 1 , !           

n

n

m T r T

t n

m T

e Ke g

nt

where

 

 

 

2 1 log 2 , n S

r T t

K f t t                       

 

 

 

2 1 2 , n

r T t

K log S

g t t

                  v         

is the initial value of

 

S TK

max

S T

 

K, 0 .

S S t

 

,

nd

r is the risk-free interes K is the strike price, a T is the expiry date. For n

t rate,

give N t

 

n, we have rn rmnlog 1

m T

and the 

 

t , variance 2

 process has mean 

f t

, skewness , an

rem is given i

d kur n the follo

n Thava ngh 4].

erentiable function tosis neswara     n a .The nd Si [

wing theo

and g x

 

Theorem 2.3 For any twice continuously diff f x

 

, the call price is given by

 

 

 

 

 

 

 

 

 

 

 

 

 

1 1 1 e e !

m T m T

t n n n S f n t          

   

 

1 2

1 2 2 2 2

1 1 2 1 2 e ! 1

e ( )

! 2 1 e e ! 2 1 1 1 4 1 1 n n n r T t m T n n r T m T n K g n

S f t f t t t

n

K g t g

n

m T m T

t m T t m T                                                                     

       

 

 

 

 

 

, n

C S T

 

 

 

 

 

 

 

 

 

2

2 2 2 2

2

2 1

( 4 1)

2 1 2

!

n

r T y

n

t E t t t

t

K g t g

T n                                

 

 2 2

2 2 2

1 1 3 1 1 3 y t t t               1 2 1 1 1 e ! e e 1 1 n m T n m T n

S f t f

n m T m                      

                    here         w  

 

 

 

 

4

t t        2 2 [ ] t t              

is the kurtosis of volatility process 

 

t ,

 

 

 

4

2 2

y y t

y t          

is the kurtosis of the observed log-return y t

 

,

 

 

2 2 2 n K t                2 1 log , ( ) S

r T t

f t                

 

 

 

2 2 2 1 log 2 , n S

r T t

(8)

H. GONG ET AL. 70

 

 

 

 

 

 

 

 

 

2

2 2 2

2

2 2

2 2

2

2 2 2

2 4

1

8

2 4

2 6

exp 8

log log

log log

n n

n

t t

f t

t

t t

t

t t

S S

r T r T

K K

S S

r T

K K

   

   

  

 

 

 

     

        

 

 

 

     

   

  

   

 

  

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

2 2

2

2

2 2 2

2

2 2

2 2

2

, 8

2 ( ) 4

1

2 4

6

log log

log

n

n n

n

t

t

t t

g t

t

t t

t

r T

S S

r T r T

K K

S r T K

 

   

  

  

 

 

      

     

 

 

   

 

 

     

   

 

 

 

   

   

  

   

 

 

 

 

 

 

 

 

2 2

2

2 2 2

2

exp .

8 8

log n t

t

t t

S r T

K  

 

 

  

 

     

    

 

 

 

 

Proof of the theorem follows from Gong, Thavanes- wara nd Singh [4]. een d onstrated for Black-Scholes model with GARCH volatility and Black- Scholes model with stochastic volatility in Gong, Tha- vaneswaran and Singh [1,2]. More details of related re- Cao and Guo [31,32].

3. Conclusions

Recently Gong, Thavaneswaran and Singh [1,2] have de- monstrated the superiority of the truncated lognormal dis- tribution method for option pricing by carrying out ex- tensive empirical analysis of the European call option valuation for S & P 100 index and showing that the pro- posed method outperforms other compelling stochastic volatility pricing models. In this paper, option pricing is discussed for Black-Scholes models, stochastic volatility models, pure jump process models and jump diffusion process models. Option pricing using PDE method to- gether with FFT and the method based on a truncated lognormal distribution for Black-Scholes process and jump diffusion process are also discussed in some detail.

4. References

[1] H. Gong, A. Thavaneswaran and J. Singh, “A Black- Scholes Model with GARCH Volatility,” The Mathe-matical Scientist, Vol. 35, No. 1, 2010, pp. 37-42. [2]

Volatility Models with Application in Option Pricing,”

Journal of Statistical Theory and Practice, Vol. 4, No. 4, 2010.

[3] E. Pillay and J. G. O’ Hara, “FFT Based Option Pric g under a Mean Reverting Process with Stochastic Volatil-ity and Jumps,” Journal of Computational and Applied Mathematics, Vol. 235, No. 12, 2011, pp. 3378-3384. doi:10.1016/j.cam.2010.10.024

waran and Singh [1] and Thavanes n a Superiority of the approach has b em

cent empirical results are also given in

H. Gong, A. Thavaneswaran and J. Singh, “Stochastic

in

A. Thavaneswaran and J. Singh, “Option Pricing for Jump Diffusion Model with Random Volatility,” The Journal of Risk Finance, Vol. 11, No. 5, 2010, pp. 496-507. doi:10.1108/15265941011092077

[4]

[5] F. Black and M. Scholes, “The Valuation of Option Con- tracts and a Test of Market Efficiency,” Journal of Fi- nance, Vol. 27, No. 2, 1972, pp. 99-417.

doi:10.2307/2978484

[6] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062 [7] R. C. Merton, “Theory of Rational Option Pricing,” Jour-

nal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183.

[8] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Cur-rency Options,” The Review of Financial Studies, Vol. 6, No. 2, 1992, pp. 327-343. doi:10.1093/rfs/6.2.327

[9] S. L. Heston, “Option Pricing with Infinitely Divisible Distributions,” Quantitative Finance, Vol. 4, No. 1, 2004, pp. 515-524. doi:10.1080/14697680400000035

(9)

of Altemative Option Pricing Models,” Journal of Finance, Vol. 52, No. 5, 1997, pp. 2003-2049.

doi:10.2307/2329472

[12] L. O. Scott, “Option Pricing When the Variance Changes Randomly: Theory, Estimation, and an Application,” Jour- nal of Financial and Quantitative Analysis, Vol. 22, No. 4, 1987, pp. 419-438. doi:10.2307/2330793

hite, “The Pricing of Options on Assets Volatilities,” Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.2307/2328253

[13] J. Hull and A. W with Stochastic

[14] J. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Journal of In- ternational Economics, Vol. 24, No. 4, 1988, pp. 129- 145.

[15] P. Ritchken and R. Trevor, “Pricing Options under Gen-eralized GARCH and Stochastic Volatility Processes,”

Journal of Finance, Vol. 59, No. 1, 1999, pp. 377-402. doi:10.1111/0022-1082.00109

[16] J. B. Wiggins, “Option Values under Stochastic Volatil-ities,” Journal of Financial Economics, Vol. 19, 1987, pp. 351-372. doi:10.1016/0304-405X(87)90009-2

[17] S. L. Heston and S. Nandi, “A Closed-Form GARCH Option Valuation Model,” The Review of Financial ies, Vol. 13, No. 1, 2000, pp. 585-625.

Stud-s/13.3.585 doi:10.1093/rf

. K. Siu and L. Chan, “Option Pricing

[18] R. J. Elliot, T for

GARCH Models with Markov Switching,” International Journal of Theoretical and Applied Finance, Vol. 9, No. 6, 2006, pp. 825-841. doi:10.1142/S0219024906003846 [19] P. Christoffersen, S. Heston and K. Jacobs, “Option Val-

uation with Conditional Skewness,” Journal of Eco- no- mtrics, Vol. 131, No. 1-2, 2006, pp. 253-284.

[20] L. Mercuri, “Option Pricing in a GARCH Model with Tempered Stable Innovations,” Finance Research Letter, Vol. 5, No. 3, 2008, pp. 172-182.

doi:10.1016/j.frl.2008.05.003

[21] A. M. Badescu and R. J. Kulperger, “GARCH Optio Oricing: a Semiparametric Appr Insurance

Mathe-n oach,”

matics andEconomics, Vol. 43, No. 1, 2008, pp. 69-84. doi:10.1016/j.insmatheco.2007.09.011

[22] G. Barone-Adesi, H. Rasmussen and C. Ravanelli, “An Option Pricing Formula for the GARCH Diffusion Model,”

Computational Statistics and Data Analysis, Vol. 49, No. 2, 2005, pp. 287-310. doi:10.1016/j.csda.2004.05.014 [23] G. Barone-Adesi, R. F. Engle and L. Mancini, “A

GARCH Option Pricing Model with Filtered Historical Simulation,” Review of Financial Studies, Vol. 21, No. 3, 2008, pp. 1223-1258. doi:10.1093/rfs/hhn031

[24] A. Thavaneswaran, S. Peiris and J. Sin Kurtosis and Option P

gh, “Derivation of ricing Formulas for Popular Vola- tility Models with Applications in Finance,” Communica-tions in Statistics.Theory and Methods, Vol. 37, 2008, pp. 1799-1814. doi:10.1080/03610920701826435

[25] A. Thavaneswaran, J. Singh and S. S. Appadoo, “Option Pricing for Some Stochastic Volatility Models,” Journal of Risk Finance, Vol. 7, No. 4, 2006, pp. 425-445. doi:10.1108/15265940610688982

[26] S. J. Taylor, “Asset Price Dynamics, Volatility, and Pre- diction,” Princeton University Press, Princeton, 2005. [27] J. M. Steele, “Stochastic Calculus and Financial Applica-

tions,” Springer, New York, 2001.

tility,” European Journal

[28] J. Vecer, “Stochastic Finance: A Numeraire Approach,” Chapman & Hall/CRC Press, London, 2011.

[29] H. Y. Wong and Y. W. Lo, “Option Pricing with Mean Reversion and Stochastic Vola

of Operational Research, Vol. 197, No. 1, 2009, pp. 179- 187. doi:10.1016/j.ejor.2008.05.014

[30] D. Duffie, J. Pan and K. Singleton, “Transform Analysis and Asset Pricing for Affine Jump-Diffusion,” Econo- metrica, Vol. 68, No. 6, 2000, pp. 1343-1376.

doi:10.1111/1468-0262.00164

[31] L. Cao and Z. F. Guo, “Applying Gradient Estimation

Financial

Technique to Estimate Gradients of European Call fol-lowing Variance-Gamma,” Proceedings of Global Con-ference on Business and Finance, Vol. 6, No. 2, 2011, pp. 12-18.

[32] L. Cao and Z. F. Guo, “Applying Variance Gamma Cor-related to Estimate Optimal Portfolio of Variance Swap,”

Figure

Updating...