A GEOMETRIC BROWNIAN MOTION MODEL WITH COMPOUND …science.sut.ac.th/mathematics/finance/images/pdf/paper/... · 2013-02-10 · A GEOMETRIC BROWNIAN MOTION MODEL … 27 This process

Advances and Applications in Statistics Volume 16, Number 1, 2010, Pages 25-47 This paper is available online at http://pphmj.com/journals/adas.htm© 2010 Pushpa Publishing House

:tionClassificaject Sub sMathematic 2010 60G22. Keywords and phrases: geometric Brownian motion, compound Poisson process, fractional

stochastic volatility, approximate models.

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

*Corresponding author

Received April 21, 2010

A GEOMETRIC BROWNIAN MOTION MODEL WITH COMPOUND POISSON PROCESS AND FRACTIONAL STOCHASTIC VOLATILITY

A. INTARASIT and P. SATTAYATHAM*

Department of Mathematics Suranaree University of Technology Nakhon Ratchasima, Thailand e-mail: [email protected]

Abstract

In this paper, we introduce an approximate approach to a geometric Brownian motion (gBm) model with compound Poisson processes and fractional stochastic volatility. Based on a fundamental result on the

2L -approximation of this fractional noise by semimartingales, we prove a convergence theorem concerning an approximate solution. A simulation example shows a significant reduction of error in a gBm with jump and fractional stochastic volatility as compared to the stochastic volatility.

1. Introduction

Let ( )P,, ℑΩ be a probability space and all the processes that we shall consider

in this paper will be defined in this space. Then a geometric Brownian motion model (or a diffusion model) is a model of the form

A. INTARASIT and P. SATTAYATHAM

26

( ),ttt dWdtSdS σ+μ= [ ]Tt ,0∈ and ,∞<T (1)

where ,ℜ∈μ ,0>σ ( ) [ ]TttSS ,0∈= is a process representing the price of the

underlying assets, and ( ) [ ]TttW ,0∈ is the standard Brownian motion.

Over the last decade, many academic researchers have tried to extend and improve the classical geometric Brownian motion model in various directions. Some researchers represent rare events by jumps and introduce a model of jump diffusion (see Merton [1] and Kou [2]). Other authors try to provide a more realistic stochastic process for the underlying process (e.g., stock price) by introducing a stochastic process for the volatility, i.e., with the variance of the stock return as random. See, for example, Hull and White [3], Stein and Stein [4] and Heston [5]. Since there is an empirical study showing that the behavior of stock price exhibits a long-range dependence, Thao [6] replaced Brownian motions (Bm) by fractional Brownian motions (fBm) in the diffusion model. Moreover, Sattayatham et al. [7] extended Thao’s results by adding a Poisson jump into the model. In this paper, we shall extend our investigations by replacing a Poisson jump by a compound Poisson jump and assuming that the variance of the stock return follows a fractional stochastic process.

The paper is organized as follows: In Section 2, we recall some preliminary properties of an approximate approach to fBm. A jump measure in the compound Poisson process is reviewed in Section 3. In Section 4, the geometric Brownian motion model with a compound Poisson process and fractional stochastic volatility is introduced. The convergent theorem of the approximate solution to the limit process is established in Section 5. Finally, we give some simulation examples to show the accuracy of the approximations by the geometric Brownian motion with stochastic volatility as compared to the geometric Brownian motion with fractional stochastic volatility.

2. An Approximate Approach to fBm

In this section, we shall review definitions of a fBm and its approximation by a semimartingale. Details of the discussion can be found in Thao [6].

A fractional Brownian motion with Hurst ( )1,0∈H is a centered Gaussian

process ( ) 0≥= tHt

H BB with zero mean, and the covariance function is given by

( ) ( ) ( ).21, 222 HHHH

sHt sttsBBEstR −−+==

A GEOMETRIC BROWNIAN MOTION MODEL … 27

This process was introduced by Kolmogorov [8] and studied further by Mandelbrot and Van Ness [9].

If ,21=H then ( ) ( )ststR ,min, = corresponds to the ordinary Brownian

motion. The fractional Brownian motion HB is neither a semimartingale nor a Markov process in the case .21≠H

The fractional Brownian motion HB exhibits a long-range dependence in the

sense that the infinite series of [ ( )]Hn

Hn

Hn BBBE −=ρ +11 is either divergent or

convergent with very late rate.

Furthermore, it has a stationary increment, i.e., the increment of HB has a normal distribution with zero mean and its variance is given by

[( ) ] ,22 HHs

Ht stBBE −=−

in an interval [ ]., ts

It is well known that HB admits the following representation:

( ) ( ) ,11

0 ⎭⎬⎫

⎩⎨⎧

−+α−Γ

= ∫ αt

stHt dWstZB

where ( ) 0≥= ssWW is a standard Brownian motion ,21−=α H ( )1,0∈H and

[( ) ( ) ]∫ ∞−

αα −−=0

.st dWsstZ

Instead of ,HB Alòs et al. [10] proposed using

( )∫ α−=t

st dWstB0

(2)

since the process tZ has absolutely continuous trajectories, so it suffices to consider

only the term ,tB that has a long-range dependence. Note that tB can be

approximated by

( )∫ αε ε+−=t

st dWstB0

(3)


28

in the sense that εtB converges to tB in ( )ΩpL as 0→ε for any ,2≥p uniform

with respect to [ ]Tt ,0∈ (see, Dung [11]). Since ( ) [ ]TttB ,0∈ε is a continuous

semimartingale, Ito calculus can be applied to the following SDE:

( ),εεε σ+μ= ttt dBdtSdS .0 Tt ≤≤

Let εtS be the solution of the above equation. Because of the convergence of ε

tB to

tB in ( )Ω2L when ,0→ε we shall define a solution of a fractional stochastic

differential equation of the form

( ),ttt dBdtSdS σ+μ= ,0 Tt ≤≤

to be a process ∗tS defined on the probability space ( )P,, ℑΩ such that the process

εtS converges to ∗

tS in ( )Ω2L as 0→ε and the convergence is uniform with

respect to [ ].,0 Tt ∈ This definition will be applied to the other similar fractional

stochastic differential equations which will appear later.

3. Compound Poisson Processes

In this section, we shall review the notion of the compound Poisson process and some of its properties which will be useful in the sequel; see Cont and Tankov [12, pp. 57-78] for the details.

Let ( )P,, ℑΩ be a probability space, with dE ℜ⊂ and μ as a (positive) Radon

measure on ( )., EE Then a Poisson random measure on E with intensity measure μ

is an integer-valued random measure NE →×Ω:M such that:

• For (almost all) ,Ω∈ω ( )⋅ω,M is an integer-valued Radon measure on E

and, for any bounded measurable set ,EA ⊂ ( ) ( ) ∞<=⋅ AMAM :, is an integer-

valued random variable.

• For each measurable set ,EA ⊂ ( )AM is a Poisson random variable with

parameter ( )Aμ such that

,N∈∀k ( )( ) ( ) ( )( ) .!kAekAMP

kA μ== μ−


• For disjoint measurable sets ,...,,1 E⊂nAA the random variables ( ),1AM

( )nAM..., are independent.

We can prove that for any Radon measure μ on ,dE ℜ⊂ there exists a Poisson

random measure M on E with intensity μ. Consequently, any Poisson random measure on E can be represented as a counting measure associated with a random sequence of points in E, i.e., there exists ( )( ) ,1≥ω nnT such that

,E∈∀A ( ) ( )( ) ( ){ }∑≥

∈ω≥=ω=ω1

.,11,n

inA ATiTAM # (4)

Define a random variable ,1∑=τ=

n

iinT where ( ) 1≥τ ii is a sequence of independent

exponential random variables with parameter λ, that is, ( ) .ti etP λ−=>τ The process

( ) ,0≥ttN defined by

∑≥

>=1

1n

Ttt nN

is called a Poisson process with intensity λ.

Moreover, by equation (4), the Poisson process may be expressed in terms of the Poisson random measure M in the following way:

( ) [ ]( ) ( )[ ]∫ ω=ω=ω

tt dsMtMN

,0,,,0,

where ds is the Lebesgue area element on [ ].,0 t

A compound Poisson process on dℜ with intensity 0>λ and jump size distribution f is a stochastic process tX defined as

∑=

=tN

iit YX

1,

where jump sizes iY are independent and identically distributed (i.i.d.) with

distribution f and ( ) 0≥ttN is a Poisson process with intensity λ, independent from

( ) .1≥iiY The Poisson process itself can be seen as a compound Poisson process on

ℜ such that .1≡iY This explains the origin of the term “compound Poisson” in the definition.


30

For every compound Poisson process ( ) 0≥ttX on dℜ with intensity λ and jump

size distribution f, its jump measure

( ) ( ){ }BtXXBJ ttX ∈−= − ,#

is a Poisson random measure on [ )∞× ,0dℜ with intensity measure ( ) =×μ dtdx

( ) ( ) ,dtdxfdtdx λ=ν where B is a measurable subset of [ )∞× ,0dℜ and ν is the

Lévy measure of the compound Poisson process.

This fact implies that every compound Poisson process can be represented in the following form:

( )[ ][ ]

∑ ∫∈ ×

×=Δ=ts t

Xst ddsdxxJXX

,0 ,0,

ℜ

where XJ is a Poisson random measure with intensity measure ( ) .dtdxν

Let E be a measurable subset of .dℜ Then for a measurable function Ef :

[ ] ,,0 dT ℜ→× we can construct an integral with respect to the Poisson random

measure M. It is given by the random variable

( ) ( ) ( ) ( )( )[ ]∫ ∑× ≥

⋅⋅=⋅tE n

nn TYfdsdxMsyf,0 1

.,,,

4. A gBm with Compound Poisson Jumps and Fractional Stochastic Volatility

Suppose ( )P,, ℑΩ is a probability space on which we define two standard

Brownian motions ( ) 0≥ttW and ( ) 0≥ttW with correlation ρ and a compound Poisson

process ( ) 0≥ttZ with intensity λ and Gaussian distribution of jump sizes.

We assume that the σ-algebras generated, respectively, by ( ) ,0≥ttZ ( ) 0≥ttW and

( ) 0≥ttW are independent.

Suppose that a single stock price tS and its volatility 2ttv σ= satisfy the

following stochastic differential equations:

( ) ,tttttt dZSdWvdtSdS−

++μ= (5)

( ) tttt Wdvdtvdv ξ+θ−ω= (6)


with initial conditions ( ) ( )Ω∈== 200 LSS tt and ( ) ( ),200 Ω∈== Lvv tt where

μ is the (deterministic) instantaneous drift of stock price returns,

ω is the mean long-term volatility,

θ is the rate at which the volatility reverts toward its long-term mean,

ξ is the volatility of the volatility process.

The notation −tS means that whenever there is a jump, the value of the process

before the jump is used on the left-hand side of the formula. The last term of equation (5) is just a symbol. More precisely, it can be defined by a stochastic integral with respect to the Poisson random measure ( )⋅ω,N as the sum of jumps of

a Poisson process ,tN

∫ ∫ ∑=

−− Δ==t t N

iTsssss

t

iSdNSYdZS0 0 1

, (7)

where tY is a random jump amplitude.

We now assume that the s’iT correspond to the jump times of a Poisson process

tN and that iY is a sequence of identically distributed random variables with values

in ( ).,1 ∞− Let tS be a predictable process. Then at time ,iT the jump of the

dynamics of tS is given by

( ) ( ) ( )−− =−=Δ iiiiT TSYTSTSS i :

which, by the assumption ,1−>iY leads always to positive values of the prices.

To solve equation (5), let us rewrite it into an integral form as follows:

∫ ∫ ∑=

Δ++μ+=t t N

iTsssst

t

iSdWSvdsSSS0 0 1

0 . (8)

Assume .0

2 ∞<⎥⎦⎤

⎢⎣⎡∫

Ttt dtSvE Then, by an application of Ito’s lemma for the jump-

diffusion process (Cont and Tankov [12, p. 275]) on equation (8) with ( )tSf t ,

( ),log tS= we get

( ) ,1log21loglog

0 0 00 ∫ ∫ ∫ +++−μ+=

t t tssssst dNYdWvdsvtSS


32

or, equivalently,

( ) .1log21exp

0 0 00 ⎥

⎦

⎤⎢⎣

⎡+++−μ= ∫ ∫ ∫


Since in many problems related to network traffic analysis, mathematical finance, and many other fields, the processes under study seem empirically to exhibit long-range dependent properties, their dynamics should be driven by a fractional Brownian process. Hence, instead of (6), we consider the fractional version of (6):

( ) ,ttttttt dNYSWvdtSdS −++μ= (9)

( ) ,tttt dBvdtvdv ξ+θ−ω= (10)

where tB is as given in equation (2). The corresponding approximately fractional

model can be defined, for each ,0>ε by

( ) ,ttttttt dNYSdWvdtSdS ε−

εεε ++μ= (11)

( ) ,εεεε ξ+θ−ω= tttt dBvdtvdv (12)

where εtB is as given in equation (3).

In the paper of Plienpanich et al. [14], the solution of the approximate model

(12) with initial condition ( ) ( )Ω∈=ε= 200 Lvv tt is given by

( ) ( ),expexp0

0 tBdsBsvv tt

st κ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+= εεε ∫ (13)

where ,21 22 αεξ+θ=κ ,0>ε ( ),21,0∈α and θ, ξ are real constants.

Assume that ( ) .0

2 ∞<⎥⎦⎤

⎢⎣⎡∫ εεT

tt dtSvE Using Ito’s formula for the jump process

again, the solution of the approximate model (11) is given by

( )∫ ∫ ∫ +++−μ+= εεεt t t

ssssst dNYdWvdsvtSS0 0 0

0 ,1log21loglog

or, equivalently,

( ) .1log21exp

0 0 00 ⎥

⎦

⎤⎢⎣

⎡+++−μ= ∫ ∫ ∫εεε

t t tssssst dNYdWvdsvtSS (14)


5. Convergence of a Solution of an Approximate Model

Before stating the main theorem concerning the convergence εtS to a random

variable ( )Ω∈∗2LSt as ,0→ε we first prove the convergence of the process

{ }0: >εεtv in ( )ΩrL as .0→ε Denoting the norm in ( )ΩrL by :r⋅

Lemma 1. Let [ )∞∈ ,1,, rqp satisfy .111qpr

+= Suppose that ( )⋅0v is a

random variable such that .0 ∞<pvE Then ∞<εrtv for all [ ].,0 Tt ∈

Proof. Using the fact that ,qpr gffg ≤ where [ )∞∈ ,1,, rqp and

qpr111 += (see Jones [13, p. 226]), we have

rtvε

( ) ( )r

tt

s tBdsBsv κ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+= εε∫ expexp

00

( ) ( ) ( )r

tt

srt tBdsBstBv κ−ξξ−κω+κ−ξ≤ εεε ∫ expexpexp0

0

( ) ( ) ( ) qtp

tsqtp tBdsBstBv κ−ξξ−κω+κ−ξ≤ εεε ∫ expexpexp

00

( ( ) ) ( ) ( ) qtt

psqtp tBdsBstBv κ−ξξ−κω+κ−ξ≤ εεε ∫ expexpexp0

0

( ) ( ) ( ) ( ) ( ) ( ) ( ) 22210

222222expexpexp tqtptqpp etettetvE εεε γξγξγξ κ−κω+κ−≤

,∞< for all [ ].,0 Tt ∈

For example, we shall compute ( ) .exp qt tB κ−ξ ε Note that εtB is a Gaussian

process with null means and finite variance. Let ( )t2εγ be the variance of .εtB Then

we get ( ) ( )12

121222

+αε−ε+==γ

+α+αε

εtBEt t (see Dung [11]). Hence


34

( ) qt tB κ−ξ εexp

( ) [ ] qqBteEt 1expεξκ−=

( ) [ ( )] qBq teEt 1expεξκ−=

( ) ( )

q

tz

zq

tdzeet

1

2 2

2

21exp

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

γπκ−= ∫

∞

∞−

γ

−

ξ ε

( ) ( )( ( ) ( ) )

( )

q

ttqztqz

tq

tdzeet

1

22

2 2

42222

222

21exp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γπκ−= ∫

∞

∞−

γ

γξ+ξγ−−

γξ ε

εε

ε

( ) ( )( )

( ( ))( )

q

ttqz

tq dzet

et

1

22 2

22

222

21exp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γπκ−= ∫

∞

∞−

γ

ξγ−−

ε

γξ ε

ε

ε

( ) ( ) ,exp 222∞<κ−= εγξ tqet for all [ ].,0 Tt ∈

The other expressions can be computed similarly. This proves the lemma.

Lemma 2. Let [ )∞∈ ,1r and 2, ≥qp with qpr111 += and suppose that

( )⋅0v is a random variable such that .0 ∞<pvE Then the process { }0: >εεtv

converges to tv in ( )ΩrL as .0→ε This convergence is uniform with respect to

[ ].,0 Tt ∈

Proof. Define a process [ ]{ }Ttvt ,0: ∈ as follows:

( ) ( ),expexp0

0 tBdsBsvv tt

st κ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+= ∫

where all the parameters are defined the same as in equation (13).


Using the fact that tB is a Gaussian process with null means and finite variance

and its variance ( ) ,12:12

22+α

==γ+αttBEt as in Lemma 1, we can show that ∈tv

( ).ΩrL Next we compute

( ) ( )tBdsBsvvv tt

stt κ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+=− εεε ∫ expexp

00

( ) ( )tBdsBsv tt

s κ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+− ∫ expexp

00

[ ( ) ( )]tBtBv tt κ−ξ−κ−ξ= ε expexp0

( ) ( )

( ) ( ).

expexp

expexp

0

0

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

κ−ξξ−κ−

κ−ξξ−κω+

∫∫ εε

tts

tts

tBdsBs

tBdsBs

The second expression of the last equation is equal to

( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛κ−ξξ−κ−κ−ξξ−κω ∫ ∫εε

t ttsts tBdsBstBdsBs

0 0expexpexpexp

( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛κ−ξξ−κ−κ−ξξ−κω= ∫ ∫ εεε


0 0expexpexpexp

( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛κ−ξξ−κ−κ−ξξ−κω+ ∫ ∫ε


0 0expexpexpexp

( ) [ ( )( ) ( ( ( )) )]1expexpexp0

−−ξκ−ξ⎥⎦

⎤⎢⎣

⎡ξ−κω= εε∫ ttt

ts BBtBdsBs

( )( ) ( ( ( )) ) ( ).exp1expexp0

tBdsBBBs tt

sss κ−ξ⎥⎦

⎤⎢⎣

⎡−−ξξ−κω+ ∫ ε

Consequently,

( ) [ ( ( )) ]1expexp0 −−ξκ−ξ=− εεttttt BBtBvvv

( ) [ ( )( ) ( ( ( )) )]1expexpexp0

−−ξκ−ξ⎥⎦

⎤⎢⎣

⎡ξ−κω+ εε∫ ttt

ts BBtBdsBs

( )( ) [ ( ( )) ] ( ).exp1expexp0

tBdsBBBs tt

sss κ−ξ⎥⎦

⎤⎢⎣

⎡−−ξξ−κω+ ∫ ε


36

Hence,

rtt vv −ε

( ) [ ( ( )) ] rttt BBtBv 1expexp0 −−ξκ−ξ≤ ε

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

tttt

s BBtBdsBs 1expexpexp0

−−ξκ−ξ⎥⎦

⎤⎢⎣

⎡ξ−κω+ εε∫

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

tt

sss tBdsBBBs κ−ξ⎥⎦

⎤⎢⎣

⎡−−ξξ−κω+ ∫ ε exp1expexp

0

( ) [ ( ( )) ] qttqtp BBtBv 220 1expexp −−ξκ−ξ≤ ε

( ) ( ) ( ( ( )) ) qttqtp

ts BBtBdsBs 220

1expexpexp −−ξκ−ξξ−κω+ εε∫

( )( ) [ ( ( )) ] ( ) .exp1expexp0 qt

p

tsss tBdsBBBs κ−ξ⎥

⎦

⎤⎢⎣

⎡−−ξξ−κω+ ∫ ε (15)

We aim to prove that 0→−εrtt vv in ( )ΩrL as .0→ε To do this we note that

( ) ∞<= 2100

pp vEv and, recalling that tB is the Gaussian process with null

means and finite variance ,2tγ then

( ) ( ) [ ] ( )[ ] qBqqqBqt tt eEteEttB 212212

2 expexpexp ξξ κ−=κ−=κ−ξ

( )

qz

zq

tdzeet t

21

22 2

2

21exp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γπκ−= ∫

∞

∞−

γ

−

ξ

( )

( )q

qzqzq

tdzeet t

tt

t

21

244

2 2

22222

222

21exp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γπκ−= ∫

∞

∞−

γ

γξ+ξγ−−

γξ

( )

( )q

qz

t

q dzeet t

t

t

21

22

2 2

22

222

21exp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γπκ−= ∫

∞

∞−

γ

ξγ−−

γξ

( ) ,exp22

∞<κ−= γξ tqet for all [ ].,0 Tt ∈ (16)


Next, it follows from the relation ( )qqA

AoAe q2212 +=− that

( ( ( )) ) ( ) ( ( ( )) ).1exp 222 qttqttqtt BBoBBBB −ξ+−ξ=−−ξ εεε (17)

Since ( ) 02 →−εqtt BB as ,0→ε equation (17), and the first expression of

equation (15) approaches zero as .0→ε

For the second expression of (15), we note that

( ) ( )∫∫ εε ξ−κ≤ξ−κt

psp

ts dsBsdsBs

00expexp

( ) ( ) ,exp 222∞<κ= εγξ tpett for all [ ].,0 Tt ∈

Since (17) approaches zero as ,0→ε the second expression of equation (15)

approaches zero as .0→ε

Finally, for the third expression of (15), we have

( )( ) [ ( ( )) ]p

tsss dsBBBs∫ −−ξξ−κ ε

01expexp

( )( ) [ ( ( )) ]∫ −−ξξ−κ≤ εt

psss dsBBBs0

1expexp

( )( ) ( ( ))∫ −−ξξ−κ≤ εt

pssps dsBBBs0 22 1expexp

[ ( ) ( ( ( )) )] ( )( )∫ ξ−κ−ξ+−ξ= εεt

pspssqtt dsBsBBoBB0 222 exp

( ) [ ( ) ( ( ( )) )] 0exp 2222

→−ξ+−ξκ≤ εεγξpssptt

p BBoBBett t

as .0→ε Thus, the third expression of equation (15) approaches zero as .0→ε

Consequently, all expressions of the right hand side of equation (15) approach

zero as .0→ε Therefore, tt vv →ε in ( )ΩrL as 0→ε and this convergence does

not depend on t, i.e., the convergence is uniform with respect to [ ].,0 Tt ∈


38

Now we ready to state and prove our main results. The solution of the approximated model (11) is given by

( ) .1log21exp

0 0 00 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+++−μ= ∫ ∫ ∫εεε


Define a stochastic process ∗tS as follows:

( ) .1log21exp

0 0 00 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+++−μ= ∫ ∫ ∫∗


The following theorem shows that the process ∗tS is the limit process of ε

tS in

( )Ω2L as .0→ε Hence, by definition, ∗tS will be the solution of equation (9).

Theorem 3. Suppose that ( )⋅0S is a random variable such that 40SE is finite

and the initial condition ( ) .00 ≠⋅v The stochastic process εtS of (18) converges to

the limit process ∗tS in ( )Ω2L as 0→ε and the convergence is uniform with respect

to [ ]Tt ,0∈ with .210 <α<

Proof. It follows from equations (18) and (19) that

( ) ( ) .121exp

0 000 ⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−−⎟⎟

⎠

⎞⎜⎜⎝

⎛=− ∫ ∫ εε

∗∗ε

t tsssss

ttt dWvvdsvvS

SSSS

Thus

2∗ε − tt SS

( ) ( )40 0040 12

1exp ⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−−⎟⎟

⎠

⎞⎜⎜⎝

⎛≤ ∫ ∫ εε

∗ t tsssss

t dWvvdsvvSSS

( ) ( ) .121exp

80 08040 ⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−−≤ ∫ ∫ εε

∗ t tsssss

t dWvvdsvvSSS (20)

The following three parts show an approximation of norm 2∗ε − tt SS in equation

(20):


• ∞<== 40

40

440 SESES by the assumptions of the theorem.

• We note that

80SSt∗

( )80 0 0

1log21exp ⎟⎟

⎠

⎞⎜⎜⎝

⎛+++−μ= ∫ ∫ ∫

t t tsssss dNYdWvdsvt

( )8000

exp21exp1logexp ⎟⎟

⎠

⎞⎜⎜⎝

⎛×⎟⎟

⎠

⎞⎜⎜⎝

⎛−×⎟⎟

⎠

⎞⎜⎜⎝

⎛++μ= ∫∫∫

tss

ts

tss dWvdsvdNYt

( )320160 2

1exp1logexp ⎟⎟⎠

⎞⎜⎜⎝

⎛−×⎟⎟

⎠

⎞⎜⎜⎝

⎛++μ≤ ∫∫

ts

tss dsvdNYt

.exp320 ⎟⎟

⎠

⎞⎜⎜⎝

⎛× ∫

tss dWv (21)

In order to get an approximation of equation (21), we first note that

( ) ( ) ( )161160

1logexpexp1logexp ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+×μ=⎟⎟

⎠

⎞⎜⎜⎝

⎛++μ ∑∫

=

tN

nn

tss YtdNYt

( ) ( )tYtN

nn μ+= ∑

=

exp1161

( ),exp TK μ≤

where K is a constant. The last inequality follows from the fact that there are a finite number of jumps in the finite interval [ ].,0 T

Moreover,

320320 21exp2

1exp ∫∫ −≤⎟⎟⎠

⎞⎜⎜⎝

⎛−

ts

ts dsvdsv

,21exp 1 ∞<⎟

⎠⎞⎜

⎝⎛≤ TM


40

where .sup: 3201 t

TtvM

≤≤= The maximum exists since ( )Ω∈ 32Lvt (by Lemma 2)

and

( ) ( )32

00

3232 expexp ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+κ−ξ= ∫

tttt dsBsvtBEv

is continuous with respect to [ ].,0 Tt ∈

For the remaining term, we note that

⎟⎟⎠

⎞⎜⎜⎝

⎛≤⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∫∫320320

expexpt

sst

ss dWvdWv

( )3202exp WWM t −≤

( ) ,exp 32 ∞<= MM

where 3202 sup: t

TtvM

≤≤= and .sup: 320

3 tTt

WM≤≤

= The maximum exists since

( ) ∞<⋅

== 1616

323232 !162

!32 tWEW tt

and is continuous with respect to [ ].,0 Tt ∈

Consequently, we see that 80S

St∗

is finite.

• The third term on the right hand side of equation (20) is calculated by using a relation ( ) ( ).1exp AoAA +=− So we have

( ) ( )80 0

121exp ⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−− ∫ ∫ εε

t tsssss dWvvdsvv

( ) ( ) RdWvvdsvvt t

sssss +−+−−≤ ∫ ∫ εε

80 021

∫ ∫ ++

−+−≤

ε

εε

t ts

ss

ssss RdW

vv

vvdsvv

0 08

882

1

,21

808

88 RWW

vv

vvvvt t

ss

ssss +−

+

−+−≤

ε

εε


where

( ) ( ) .21

80 0 ⎟⎟⎠

⎞⎜⎜⎝

⎛−+−−= ∫ ∫ εε

t tsssss dWvvdsvvoR

Hence,

( ) ( )80 0

121exp ⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+−− ∫ ∫ εε

t tsssss dWvvdsvv

,ˆ21

8

88 RM

vv

vvvvt

ss

ssss +

+

−+−≤

ε

εε (22)

where

.max:ˆ80 tTt

WM≤≤

=

Note that 80 ss vvc +≤< ε for all [ ],,0 ts ∈ since we assume that .00 ≠v

Hence, the right hand side of equation (22) approaches zero as .0→ε

Therefore ∗ε → tt SS in ( )Ω2L as .0→ε This convergence does not depend

on t and is hence uniform with respect to [ ].,0 Tt ∈

6. Simulation Examples

Let us consider the Thai stock market. Figure 1 shows the daily prices of the data set consisting of 283 open-prices of the K-BANK between July 2, 2008 and June 30, 2009. The empirical data set for these stock prices were obtained from http://www.set.or.th/. Figure 2 shows the log returns of the stock prices in that period.


42

Figure 1. Stock prices trading daily of K-BANK between July 2, 2008 and June 30, 2009.

Figure 2. Log returns on the stock prices of K-BANK between July 2, 2008 and June 30, 2009.


The statistics of stock prices and log returns are given in Table 1.

Table 1. Statistics of K-BANK data set

Stock prices Log returns

Data amount 238 237

Mean 55.2153 – 0.0001

Standard deviation 10.3166 0.0311

Skewness 0.2856 – 0.2990

Kurtosis 1.6902 4.5338

Figure 3 shows the empirical data of K-BANK open-price as compared to the price simulated by the classical gBm with compound Poisson jumps and stochastic volatility. The simulated model is

( ) ,1log21exp

0 0 00 ⎥

⎦

⎤⎢⎣

⎡+++−μ= ∫ ∫ ∫


and

( ) .tttt Wdvdtvdv ξ+θ−ω= (24)

The model parameters 125.3−=μ and .0311.0=σ The mean of =jumps

,0425.0 the standard deviation of 0.0175jumps = and the intensity .17=λ The

model parameters for stochastic volatility are ,00525.0=ω 2250.0=ξ and =θ

.000825.0 For comparative purposes, we compute the Average Relative Percentage Error (ARPE). By definition,

∑=

×−

=N

k kkk

XYX

N1

,1001ARPE

where N is the number of prices, ( ) 1≥= kkXX is the market price and ( ) 1≥= kkYY

is the model price. After working 105 trails we compute ARPE for Figure 3 which will be denoted by ARPE (3).


44

Figure 3. Price behavior of K-BANK, between July 2, 2008 and June 30, 2009, as compared with a scenario simulated from a gBm with a fractional volatility model.

bysimulated:linedasheddata,empirical:line(Solid ==

( ) ⎥⎦

⎤⎢⎣

⎡+++−μ= ∫ ∫ ∫


0 0 00 1log2

1exp

with stochastic volatility

( ) ,tttt Wdvdtvdv ξ+θ−ω=

,105=N ( ) .)23404.21:3ARPE =

Figure 4 shows the empirical data of K-BANK open-price as compared to the price that was simulated by an approximation of gBm with compound Poisson jumps and fractional stochastic volatility. The simulated model is

( ) ⎥⎦

⎤⎢⎣

⎡+++−μ= ∫ ∫ ∫εεε


0 0 00 1log2

1exp (25)

and

( ) ( ).expexp0

0 tBdsBsvv tt

tt κ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+= εεε ∫ (26)


The value of μ, σ and the parameter for jumps and volatility are the same as Figure 3. For the remaining data, we choose 00001.0=ε and .00125.0=α

Figure 4. Price behavior of K-BANK, between July 2, 2008 and June 30, 2009, as compared with a scenario simulated from a gBm with a fractional volatility model.

bysimulated:linedasheddata,empirical:line(Solid ==

( ) ⎥⎦

⎤⎢⎣

⎡+++−μ= ∫ ∫ ∫εεε


0 0 00 1log2

1exp

with stochastic volatility

( ) ( ),expexp0

0 tBdsBsvv tt

tt κ−ξ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ−κω+= εεε ∫

,105=N ,54647.19:ARPE =

where ARPE(4) is the ARPE for Figure 4.)

By comparing the ARPE of Figures 3 and 4, we can see that in case of K- BANK, the sample path of an approximation of gBm with compound Poisson jumps and fractional stochastic volatility gives a better fit with the data than the classical gBm with compound Poisson jumps and stochastic volatility.


46

Figure 5 shows the convergence of ARPE(3) and ARPE(4), working out for 25, 45, 55, 65, 75, 95 and 105 trails.

Figure 5. Convergence of ARPE(3) and ARPE(4) with 9555,45,25,=N and 105.

References

[1] R. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Econom. 3 (1976), 125-144.

[2] S. Kou, A jump-diffusion model for option pricing, Management Sci. 48 (2002), 1086-1101.

[3] J. G. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finan. 42 (1987), 281-300.

[4] E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility, Rev. Finan. Stud. 4 (1991), 727-752.

[5] S. L. Heston, A closed form solution for options with stochastic volatility with applications to bond and currency options, Rev. Finan. Stud. 6 (1993), 327-343.

[6] T. H. Thao, An approximate approach to fractional analysis for finance, Nonlinear Anal. Real World Appl. 7 (2006), 124-132.

[7] P. Sattayatham, A. Intrasit and P. Chaiyasena, A fractional Black-Scholes model with jumps, Vietnam J. Math. 35(3) (2007), 1-15.

[8] A. N. Kolmogorov, Wienersche Spiralen und einige andere nteressante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. URSS (N.S.) 26 (1940), 115-118.


[9] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422-437.

[10] El. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2, Stochastic Process. Appl. 86(1) (2000), 121-139.

[11] Nguyen Tien Dung, A class of fractional stochastic equations, Institute of Mathematics, Vietnam Academy of Science and Technology, 2007.

[12] R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, America, 2004.

[13] F. Jones, Lebesgue Integration on Euclidean Space, Jones and Bartlett Publishers, London, 1993.

[14] T. Plienpanich, P. Sattayatham and T. H. Thao, Fractional integrated GARCH diffusion limit models, J. Korean Statist. Soc. 38 (2009), 231-238.

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