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*,* umlian*33*@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.

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]

###

###

###

###

###

###

d*X t* *t X t*, d*t*( ,*t X t* )d*W t*
and

###

###

###

###

###

d *t* *r 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 x* *h 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 t* *a t X t* *b t* *t*

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

###

###

###

*V T* *h X T*

###

d . From the self-financing condition

###

###

d*V t* *a t* d*X t* *b 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*,

###

###

,*a*

*f*

*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 t* *t*

_{}

.

Because *V t*

###

is equal to both*f t X t*

###

,###

###

and###

*a t X t* *b t* *t* , 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 t* *t 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 function***f t x*

###

,###

defined by###

d_{}

_{ }

_{}

_{ }

, ,

*T*
*tV X*

*f t x* _{}*e* *h X T* *X t* *x*_{}

where _{d}_{X t}

###

_{}

_{}

###

_{t X t}_{,}

###

###

_{d}

_{t}_{}

_{}

###

_{t X t}_{,}

###

###

_{d}

_{W 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*

_{} _{}

###

###

###

###

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 x* *rxf* *t x* *x f* *t x* *rf t x*

with the terminal condition: (a) *f T x*

###

,###

*x*

*K*

###

###

###

: 0,

*h*

; (b) ; (c)

###

*T x*,

###

*h x*for some function

###

,###

*f*

*f T x* *x*; (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 x* *X 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 T* *X t* *r* *T* *t* *W T**t*

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

###

_{ }

###

###

_{ }

###

###

2_{2}

e e

e e .

*T t* *W T t*

*r T t*
*Q*

*x*

2_{2}

*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*

2_{2}

e *r T t* e *r* *T t*

*X t* *X t* *x*

*x*

*x*

_{}

In Case (d),

_{} _{}

###

_{ }

###

###

_{ }

###

_{ }

###

###

_{}

_{}

###

2_{2}

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* *t* *d N* *t*
has a solution as a geometric Poisson process

###

###

###

###

###

*t* *t*

###

###

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*

###

###

###

###

###

###

d*YX* *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* *t* *Y* exp *N* *t* *t* (2.6)
*Y*

and *X*

The values of are linked by the
rela-tionship *X* *e**Y*

, where is the size of the jump of

###

log*XY* *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*

H. GONG *ET AL*.
66

we can write

###

,###

###

###

*Y*

###

.*u* *t x* _{}*f* *X* *T* *X* *t* *x*_{}

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 _{f}X

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 t* *x*

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 x* *f* *x*
(b) The price fu sed on et

ss (2.6), *uY*

###

*t x*,

*Y*

_{t}_{}

*fY*

###

*X*

_{Y}###

*T*

###

*X*

_{Y}###

*t*

*x*

_{}satisfies

###

,###

###

, e###

###

,*Y* *Y* *Y* *Y*

*ut* *t x* *ut* *t* *x* *u* *t* *x*

###

###

e 1###

*xuYx*

###

*t x*,

###

0erminal 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 x* *f* *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

*VT*

*I*

###

*N*

###

*T*

*k*

###

*T*reference

n by

*Y*
give

###

###

###

###

###

###

###

###

###

###

###

###

###

_{}

_{}

###

###

*k*

*m*

###

! .

*Y _{T t}*

*k m*

*Y*

*e* *T**t*

*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

*Y* *t* *Y* *t*

###

*X*

*V* *t*

###

###

###

###

e _{}

###

###

_{}

###

###

.*Y _{T 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 t*2

###

are correlated with correlationcoef-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 pendeto be inde nt of both *W t*1

###

and*W t*2

###

. Furthermore,a sequence of random variables _{e}Ji_{ for }_{1} _{i}_{N t}

###

_{ is }

defined to represent the jump sizes of the Poisson
proc-ess. Each of the _{e}Ji_{’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 t* *S* *t* and applying the Itô-Doebin formula to
the two-factor mean reverting process with stochastic
volatility and jumps we have

###

###

###

###

###

###

###

###

###

1

d 2

e*J* 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*

###

2d*v* *t* *b*(*a**v* *t* )d*t* *v* *t* d*W t* , (2.8)

###

###

1 2

###

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*

*xv*

*B t T*

*C t T x*

*D*

*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* 2*k**m k**x* 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 deterministicfunctions 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 T* *C* *D 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 Pprocesses

###

###

###

###

###

1
d*X* *t* *x* d*t* *x* d*W* *t*

*X*

###

,###

###

###

2###

d*v* *t* *v* d*t* *v* d*W* *t* ,

###

ln###

,*t* *S t*

###

###

d*W* *t* d*W* *t* d*t*

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 f _{vv}* (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* *x* *v*

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}

_{e}

*i X T*

_{.}

*t* *u* *t*

_{}_{} _{}

Furthermore, if one defining the following function

###

_{; , ,}

###

_{E}

_{e}

*i X T*

_{( )}

_{,}

_{ }

_{,}

*f* *t x v* _{}_{} *X t* *x v t* *v*_{}

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

###

; , ,###

exp###

###

###

###

###

*f* *t x v* *B* *C* *x**D* *v 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*_{} *C*_{} *x**D*_{} *v f* *t 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* *t* *v t* *W t*

where d

*X t* *S t*

###

###

1 2

d*W t* d*W 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* ,

###

*x*

*v t*

###

, ###

*v*

*a*

*cv t*

###

and ###

*v*

*v t*

###

. Substitute them into (2.16), we have2

1

d 2 ln ,

1

*a* *ge*

*Ri* *i* *c*

*g*

_{} _{} _{}

###

2 d dd 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

_{ }

###

22

*v* *r* *m* *t* , *m*_{}*eY* 1_{}, and *W t*

###

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

###

*t*umps

*N t*

###

ontionary

###

_{ }

###

_{}

###

_{}

_{}

###

_{}

###

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 , 1e 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*

*n* *t*

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 T*

*K*

###

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

*r*

_{n}*r*

*m*

*n*log 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 21 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*

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
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*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
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[9] S. L. Heston, “Option Pricing with Infinitely Divisible
Distributions,” *Quantitative Finance*, Vol. 4, No. 1, 2004,
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of Altemative Option Pricing Models,” *Journal of Finance*,
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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.
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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
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[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 *
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uation with Conditional Skewness,” *Journal of Eco- no- *
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