A Geometric Brownian Motion Model with Compound Poisson …science.sut.ac.th/mathematics/pairote/uploadfiles/Artit_cmu_DG.pdf · School of Mathematics, Institute of Science Suranaree

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Outline Research Guideline Warm-Up Conceptual Research Framework

A Geometric Brownian Motion Model withCompound Poisson Process and Fractional

Stochastic Volatility

Arthit Intarasit

School of Mathematics, Institute of ScienceSuranaree University of Technology

Nakhon Ratchasima, [email protected]

February 25, 2010


Outline

1 Research Guideline

2 Warm-UpStock Price ModelLiterature Reviews

3 Conceptual Research FrameworkModel DescriptionSimulation Examples


Background

This talk is based on the manuscript

A. Intarasit and P. Sattayatham. (2010) A GeometricBrownian Motion Model with compound Poisson Process andFractional Stochastic Volatility, 18 pages.

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

This article will appear at

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Abstract

In this paper, we introduce an approximate approach to ageometric Brownian motion (gBm) model with compound Poissonprocesses and fractional stochastic volatility.

Based on a fundamental result on the L2-approximation of thisfractional noise by semimartingales, we prove a convergencetheorem concerning an approximate solution.

A simulation example shows a significant reduction of error in agBm with compound Poisson processes and fractional stochasticvolatility as compared to the classical stochastic volatility.


Research Objective

The research field of the work is in MATHEMATICALFINANCE.

The objectives of the research is to modeling STOCK PRICEspecification to STOCHASTIC VOLATILITY.

The purposed of the research:

Investigating stock price by adding a compound Poissonprocess and assuming that the variance of the stock returnfollows a fractional stochastic process.

Giving applications on K-BANK empirical data.

Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews

The Black-Scholes model

Following Black and Scholes (1973, [1]), we assume that thebehavior of the stock price (St)t≥0 of a risky assets at time t isdetermined by the following stochastic integral:

dStSt

= µdt+ σdWt, ∀t ∈ [0, T ] and T <∞, (1)

where µ ∈ < the drift rate (the average rate at which a valueincreases in a stochastic process) and (Wt)t∈[0,T ] is the standardBrownian motion (Bm). Note that the Bm Wt is called the drivingprocess of the SDE (1).

The parameter σ ∈ < is called volatility becase it characterizes thedegree of variability.

Remark: The Wiener process (or Brownian Motion) Wt is characterize by three facts: 1) W0 = 0. 2) Wt is

almost surely continuous. 3) Wt has independent increments with distribution Wt −Ws ∼ N(0, t− s) for

0 ≤ s ≤ t.


The Black-Scholes model (cont.)

If σ = 0, the equation (1) become

St = S0eµt, St=0 = S0.

i.e. it describes an investment on a non-risky asset (e.g., a bankaccount).

Applying the Ito formula (see, Lamberton and Lapeyre [2], forexample) on equation (1) with f(x) = log(x), we obtain

St = S0 exp(µt− σ2

2t+ σWt

).

This equation is called that ”a geometric Brownian motion(GBM)”.


The Black-Scholes model (cont.)

If σ = 0, the equation (1) become

St = S0eµt, St=0 = S0.

i.e. it describes an investment on a non-risky asset (e.g., a bankaccount).

Applying the Ito formula (see, Lamberton and Lapeyre [2], forexample) on equation (1) with f(x) = log(x), we obtain

St = S0 exp(µt− σ2

2t+ σWt

).

This equation is called that ”a geometric Brownian motion(GBM)”.


One-dimensional Ito’s formula

Theorem

Let the random process xt satisfies the diffusion equation

dxt = a(xt, t)dt+ b(xt, t)dWt,

where Wt is a standard Brownian motion. Let the processyt = F (xt, t) ∈ C2,1 be at least a twice differentiable function.Then the process yt satisfies Ito’s equation

dyt =(∂F∂x

a+∂F

∂t+

1

2

∂2F

∂x2b2)dt+

∂F

∂xb dWt.

(Chan and Wong [3], page 24)Remark:

Ito’s Lemma provides a derivative chain rule for stochastic functions. It gives the relationship between astochastic process and a function of that stochastic process.

A well known application in finance. When defining the returns of stock with a stochastic process, Ito’slemma is used to change the returns into stock prices.


An Extension of the Black-Scholes model

The unsatisfactory performance by the Black-Scholes model hasled to a search for better alternatives that extend the classic modelin one, or a combination, of four directions:

to allow for random jumps to occur in the stock price process,

to allow for stochastic volatility,

to allow for stochastic interest rates, and

to allow for long memory in the stock price process.

Remark:

The drift rate µ (or the interest rate r) can be assumed constant since it has little impact on short-timeoption price see Scott ([4],1997)

The process X have long memory or long-rang dependence its means that the process today may influencethe process at some time in the future or the process at long time before my influence the process today.

.








Remark:



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Remark:



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Remark:



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Remark:



.


The Merton ModelThere exist stylized facts in real stock returns that have a leptokurtic andskewed distribution while the returns in the Black-Scholes’s Model arenormally distributed.

Merton (1976, [5]) extended the Black-Scholes model by introduce anexponential Levy model:

St = S0 exp(µt+ σWt +

Nt∑i=1

Yi

)where (Nt)t∈[0,T ] is a Poisson process with intensity λ, and independentjumps Yi ∼ N(m, δ2). The Poisson process and the jumps are assumedto be independent of the Wiener process.

Sattayatham at el (2004, [6]) extended this model by introduce afractional Black-Scholes model with jumps, which the stock price havelong-memory property.

Remark: The Merton model allow to model jumps and to consider leptokurtic distributions (the distribution is

more peaked than the normal distribution with a heavier tail)


The Merton ModelThere exist stylized facts in real stock returns that have a leptokurtic andskewed distribution while the returns in the Black-Scholes’s Model arenormally distributed.

Merton (1976, [5]) extended the Black-Scholes model by introduce anexponential Levy model:

St = S0 exp(µt+ σWt +

Nt∑i=1

Yi

)where (Nt)t∈[0,T ] is a Poisson process with intensity λ, and independentjumps Yi ∼ N(m, δ2). The Poisson process and the jumps are assumedto be independent of the Wiener process.

Sattayatham at el (2004, [6]) extended this model by introduce afractional Black-Scholes model with jumps, which the stock price havelong-memory property.

Remark: The Merton model allow to model jumps and to consider leptokurtic distributions (the distribution is

more peaked than the normal distribution with a heavier tail)


The Volatility Problem

It is widely believed and experimentally verified that stocks do nothave a constant spot volatility, rather this parameter varies withtime, e.g. Hull and White (1987, [7]), Danilo and Spokoinyi (2004,[8]), Goldentyer, Klebaner and Liptser (2005, [9]).

There are three commonly used parameterizations of volatility:

Constant volatility,

Stochastic Volatility (SV) and

GARCH.


The Volatility Problem

It is widely believed and experimentally verified that stocks do nothave a constant spot volatility, rather this parameter varies withtime, e.g. Hull and White (1987, [7]), Danilo and Spokoinyi (2004,[8]), Goldentyer, Klebaner and Liptser (2005, [9]).

There are three commonly used parameterizations of volatility:

Constant volatility,

Stochastic Volatility (SV) and

GARCH.


The Heston Model (the most popular SV)

Heston (1993, [10]) considered for the stock price a stochasticvolatility model:

dStSt

= µdt+√vtdWt

where the volatility process is modeled by a square-root process:

dvt = θ(ω − vt)dt+ ξ√vtdW t

Here the processes(Wt

)t∈[0,T ] and

(W t

)t∈[0,T ] are correlated

Wiener processes:

Cov(Wt,W t

)= ρt.

Remark: The idea behind Heston’s model is he employed the square root diffusion to model the evolution of

instantaneous variance dynamics which were first used by Cox and Ross (1985, [11]) in the area of interest rate

modeling.


Stochastic Volatility models (SV)

This basic model with constant volatility σ is the starting point fornon-stochastic volatility models such as Black-Scholes andCox-Ross-Rubinstein binomial model.

For a stochastic volatility model, replace the constant volatility σ with afunction vt, that models the variance of St. This variance function is alsomodeled as brownian motion, and the form of vt depends on theparticular SV model under study.

The generalizations of the stochastic volatility models is

dSt = µStdt+√vtStdWt,

dvt = α(St, t)dt+ β(St, t)dW t

where α(St, t) and β(St, t) are some function of vt and W t is another

standard gaussian that is correlated with dWt with constant correlation

fator .


Local Volatility Models

Consider for the stock price diffusion model:

dSt = µStdt+ σ(St, t)StdWt

where the function σ determines the volatility at time t and pricelevel St. These models are called local volatility model.


GARCH model in Continous Time

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH)model is another popular model for estimating stochastic volatility.It assumes that the randomness of the variance process varies with thevariance, as opposed to the square root of the variance as in the Hestonmodel.

The standard GARCH model has the following form for the variancedifferential:

dSt = µStdt+ vtStdWt,

dvt = θ(ω − vt)dt+ ξvtdW t

where ω is the mean long-term volatility, θ is the rate at which thevolatility revert toward its long-term mean, ξ is the volatility of thevolatility process. (Nelson 1990, [12])

Remark: The GARCH model has been extended via numerous variants, including the NGARCH, LGARCH,

EGARCH, GJR-GARCH, etc.


The FIGARCH model

That volatility exhibits long memory is well established in therecent empirical literature. For example see Baillie et al. (1996,[13]) Robinson (2001, [14]), and Andersen, Bollerslev, Diebold andLabys (2003, [15]).

Baillie et al. (1996, [13]) suggest the FIGARCH model in discretetime to capture the long memory present in volatility.

For recent works in FIGARCH, Plienpanich et al (2009, [16])introduced a fractional integrated GARCH model (FIGARCH) incontinuous time, that process of the form:

dvt = (ω − θvt)dt+ ξvtdWHt

where WHt is a fractional Brownian motion (fBm).


The Bate model

Merton’s and Heston’s approaches were combined by Bates (1996,[17]), who proposed a stock price model with stochastic volatilityand jumps:

dStSt

= µdt+√vtdWt + dZt,

dvt = θ(ω − vt)dt+ ξ√vtdW t,

andCov

(Wt,W t

)= ρt.

The combination of Bates’s (gBm with compound Poissonprocess) and FIGARCH’s model still open topic !!


The Bate model

Merton’s and Heston’s approaches were combined by Bates (1996,[17]), who proposed a stock price model with stochastic volatilityand jumps:

dStSt

= µdt+√vtdWt + dZt,

dvt = θ(ω − vt)dt+ ξ√vtdW t,

andCov

(Wt,W t

)= ρt.

The combination of Bates’s (gBm with compound Poissonprocess) and FIGARCH’s model still open topic !!

Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples

Model Setting

In this research we introduce a gBm with compound Poisson jumpsand fraction stochastic volatility for modeling a stock price.

Suppose that a single stock price St and its volatility vt = σ2tsatisfy the following stochastic differential equations:

dSt = St(µdt+

√vtdWt

)+ St−dZt, (2)

dvt = (ω − θvt)dt+ ξvtdBt, (3)

with initial condition St(t=0) = S0, vt(t=0) = v0 and Bt is a fBm.

A parameter µ, ω, θ, and ξ define is as given above. The notationSt− means that whenever there is a jump, the value of the processbefore jump is used on the left-hand side of the formula.


An Approximation Approach to fBm

The Ito calculus is well-defined for an integral with respect tosemimartingale. Since the fBm is not semimartingale, hence theIto calculus cannot be applied. To overcome this problem we usean approximation approach first proposed by Thao (2006, [18]).

Remark:

A real valued process X defined on the filtered probability space (Ω,F, (Ft)t≥0, P) is calledsemimartingal if it can be decomposed as Xt = Mt + At where Mt is a local martingale and At is acadlag adapted process of locally bounded variation.

Every Ito process, cadlag process, Levy process, and Bm are semimartingal while fBm with Hurstparameter H 6= 1/2 is not a semimartingale.

The semimartingales form the largest class of processes for which the Ito integral can be defined.


An Approximation Approach to fBm (cont.)

The representation of fBm is

Bt =

∫ t

0(t− s)αdWs,

where α = H − 1/2, H ∈ (0, 1). Thao approximate Bt by

Bεt =

∫ t

0(t− s+ ε)αdWs,

in the sense that Bεt converges to Bt in Lp(Ω) as ε→ 0 for any

p ≥ 2, uniform with respect to t ∈ [0, T ].

Moreover, one can show that Bεt is a semimartingale and has a

long-range dependence in sense that∞∑n=1

E[WH1 (WH

n+1 −WHn )] =∞.


An Approximation Approach to fBm (cont.)

The representation of fBm is

Bt =

∫ t

0(t− s)αdWs,

where α = H − 1/2, H ∈ (0, 1). Thao approximate Bt by

Bεt =

∫ t

0(t− s+ ε)αdWs,

in the sense that Bεt converges to Bt in Lp(Ω) as ε→ 0 for any

p ≥ 2, uniform with respect to t ∈ [0, T ].

Moreover, one can show that Bεt is a semimartingale and has a

long-range dependence in sense that∞∑n=1

E[WH1 (WH

n+1 −WHn )] =∞.


The Solution of The Approximately Fractional Model

The corresponding approximately fractional model of equation (2) and(3) can be define, for each ε > 0, by

dSεt = Sεt(µdt+

√vεt dWt

)+ Sεt−dZt, (4)

dvεt = (ω − θvεt )dt+ ξvεt dBεt , (5)

where Bεt is as given above.

Assume that E[∫ T0vεt (St

ε)2dt] <∞. Using Ito’s formula for the jumpprocess, the solution of the the approximate model (4) is given by

Sεt = S0 exp[µt− 1

2

∫ t

0

vεsds+

∫ t

0

√vεsdWs +

∫ t

0

log(1 + Ys)dNs]

with initial condition St(t=0) = S0 ∈ L2(Ω) and the solution of equation(5) is

vεt =(v0 + ω

∫ t

0

exp(κs− ξBεs)ds)

exp(ξBεt − κt)

with initial condition vt(t=0) = v0 ∈ L2(Ω), κ = θ + 12ξ

2ε2, ε > 0,

α ∈ (0, 1/2), and θ, ξ are real constants.


The Solution of The Approximately Fractional Model

The corresponding approximately fractional model of equation (2) and(3) can be define, for each ε > 0, by

dSεt = Sεt(µdt+

√vεt dWt

)+ Sεt−dZt, (4)

dvεt = (ω − θvεt )dt+ ξvεt dBεt , (5)

where Bεt is as given above.

Assume that E[∫ T0vεt (St

ε)2dt] <∞. Using Ito’s formula for the jumpprocess, the solution of the the approximate model (4) is given by

Sεt = S0 exp[µt− 1

2

∫ t

0

vεsds+

∫ t

0

√vεsdWs +

∫ t

0

log(1 + Ys)dNs]

with initial condition St(t=0) = S0 ∈ L2(Ω) and the solution of equation(5) is

vεt =(v0 + ω

∫ t

0

exp(κs− ξBεs)ds)

exp(ξBεt − κt)

with initial condition vt(t=0) = v0 ∈ L2(Ω), κ = θ + 12ξ

2ε2, ε > 0,

α ∈ (0, 1/2), and θ, ξ are real constants.


The Solution of The Fractional SDE Model

Sinc (Bt)t∈[0,T ] is a continuous semimartingale then Ito calculus can beapplied to the following SDE

dXεt = Xε

t (µdt+ σdBεt ), 0 ≤ t ≤ T.

Let Xεt be the solution of the above equation. Because of the

convergence of Bεt to Bt in L2(Ω) when ε→ 0, we shall define a solutionof a fractional stochastic differential equation of the form

dXt = Xt(µdt+ σdBt), 0 ≤ t ≤ T,

to be a process X∗t defined on the probability space (Ω,F ,P) such thatthe process Xε

t converges to X∗t in L2(Ω) as ε→ 0 and the convergenceis uniform with respecto t ∈ [0, T ].

This definition will be applied to the other similar fractional stochastic

differential equations which will appear later.


Convergence of Solution of An Approximate Model

Define a stochastic process S∗t as follows:

S∗t = S0 exp[µt− 1

2

∫ t

0vsds+

∫ t

0

√vsdWs +

∫ t

0log(1 + Ys)dNs

](6)

If we can show that the process S∗t is the limit process of Sεt inL2(Ω) as ε→ 0. Hence, by definition, S∗t will be the solution ofequation (2).


The Main Theorem

Theorem

Suppose that S0(·) is a random variable such that E|S0|4 is finiteand the initial condition v0(·) 6= 0. The stochastic process Sεt ofequation (4) converges to the limit process S∗t define by (6) inL2(Ω) as ε→ 0 and the convergence is uniform with respect tot ∈ [0, T ] with 0 < α < 1/2.

The technique are follow:

Using the fact that ||fg||r ≤ ||f ||p||g||q where p, q, r ∈ [1,∞) and1r = 1

p + 1q (see Jones [19], page 226), we compute

||Sεt − S∗t ||2 = ||S0||4∣∣∣∣∣∣∣∣S∗tS0

∣∣∣∣∣∣∣∣8∣∣∣∣∣∣∣∣[exp

(− 1

2

∫ t

0

(vεs − vs)ds+

∫ t

0

(√vεs −

√vsds

)− 1

]∣∣∣∣∣∣∣∣8

(7)


The Main Theorem

Theorem

Suppose that S0(·) is a random variable such that E|S0|4 is finiteand the initial condition v0(·) 6= 0. The stochastic process Sεt ofequation (4) converges to the limit process S∗t define by (6) inL2(Ω) as ε→ 0 and the convergence is uniform with respect tot ∈ [0, T ] with 0 < α < 1/2.

The technique are follow:

Using the fact that ||fg||r ≤ ||f ||p||g||q where p, q, r ∈ [1,∞) and1r = 1

p + 1q (see Jones [19], page 226), we compute

||Sεt − S∗t ||2 = ||S0||4∣∣∣∣∣∣∣∣S∗tS0

∣∣∣∣∣∣∣∣8∣∣∣∣∣∣∣∣[exp

(− 1

2

∫ t

0

(vεs − vs)ds+

∫ t

0

(√vεs −

√vsds

)− 1

]∣∣∣∣∣∣∣∣8

(7)


The Technique of Proof

Using a relation exp(A)− 1 = A+ o(A), the third term of equation (7)become

|∣∣∣∣[exp

(− 1

2

∫ t

0

(vεs − vs)ds+

∫ t

0

(√vεs −

√vsds

)− 1

]∣∣∣∣∣∣∣∣8

=1

2||vεs − vs||8 +

||vεs − vs||8||√vεs −

√vs||8

M +R (8)

where R =∣∣∣∣∣∣o(− 1

2

∫ t0(vεs − vs)ds+

∫ t0(√vεs −

√vsds

)− 1∣∣∣∣∣∣8

and

M := ||Wt −W0||8

Note that 0 < c ≤ ||vεs − vs||8 for all s ∈ [0, t] since we assume thatv0 > 0. If we can proof that vεs → vs in L8(Ω), the third term ofequation (7) approaches zero as ε→ 0.

Since the first and second term are finite, therefore Sεt → S∗t in L2(Ω) as

ε→ 0. And this convergence does not depend on t and is hence uniform

with respect to t ∈ [0, T ].




|∣∣∣∣[exp

(− 1

2

∫ t

0

(vεs − vs)ds+

∫ t

0

(√vεs −

√vsds

)− 1

]∣∣∣∣∣∣∣∣8

=1

2||vεs − vs||8 +

||vεs − vs||8||√vεs −

√vs||8

M +R (8)

where R =∣∣∣∣∣∣o(− 1

2


∫ t0(√vεs −

√vsds

)− 1∣∣∣∣∣∣8

and

M := ||Wt −W0||8








|∣∣∣∣[exp

(− 1

2

∫ t

0

(vεs − vs)ds+

∫ t

0

(√vεs −

√vsds

)− 1

]∣∣∣∣∣∣∣∣8

=1

2||vεs − vs||8 +

||vεs − vs||8||√vεs −

√vs||8

M +R (8)

where R =∣∣∣∣∣∣o(− 1

2


∫ t0(√vεs −

√vsds

)− 1∣∣∣∣∣∣8

and

M := ||Wt −W0||8






The Convergent Lemma of Volatility

The convergent of approximate fSV proved by the following twolemmas:

Lemma (1)

Let p, q, r ∈ [1,∞) satisfy 1r = 1

p + 1q . Suppose that v0(·) is

random variable such that E|v0|p <∞ then ||vεt ||r <∞ for allt ∈ [0, T ].

Lemma (2)

Let r ∈ [1,∞) and p, q,≥ 2 satisfy 1r = 1

p + 1q and suppose that

v0(·) is random variable such that E|v0|p <∞. Then the processvεt : ε > 0 converges to vt in Lr(Ω) as ε→ 0. This convergenceis uniform with respect to t ∈ [0, T ].




Lemma (1)




Lemma (2)







Lemma (1)




Lemma (2)





Simulation Examples

Figure1: Stock prices trading daily of K-BANK between July 2, 2008 andJune 30, 2009, which obtained from http://www.set.or.th/.


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


Setting µ = −3.125, σ = 0.311. The mean of jumps =0.0425, the sd of jumps =0.0175 and the intensityλ = 17. The model parameters for stochastic volatility are ω = 0.00525, ξ = 0.2250, and θ = 0.000825.

Figure 3: Price behavior of K-BANK between July 2, 2008 and June 30, 2009, as compared with a scenariosimulated from a classical gBm with a volatility model. (solid line:=empirical data,red dashed line:=simulated by

St = S0

[µt−

1

2

∫ t

0vsds +

∫ t

0

√vsdWs +

∫ t

0log(1 + Ys)dNs

]

with stochastic volatility dvt = (ω − θvt)dt + ξvtdW t,

N = 105, ARPE(3) = 21.23404

where ARPE(3) is the ARPE for Figure 3. Note that ARPE:= 1N

∑Nk=1

|Xk−Yk|Xk

× 100, where N is the

number of prices, (Xk)k≥1 is the market price and (Yk)k≥1 is the model price)


Figure 4: Price behavior of K-BANK between July 2, 2008 and June 30, 2009, as compared with a scenariosimulated from a gBm with a fractional volatility model. (solid line:=empirical data, blue dashed line:=simulated by

Sεt = S0

[µt−

1

2

∫ t

0vεsds +

∫ t

0

√vεsdWs +

∫ t

0log(1 + Ys)dNs

]with fractional stochastic volatility

dvεt =

(v0 + ω

∫ t

0exp(κ− ξBε

t )ds)

exp(ξBεt − t)

N = 105, ARPE(4) = 19.54647.

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


Fig 5. Convergence of ARPE(3) and ARPE(4) with N = 25, 45, 55, 96

and 105.


Research Possibility

Extend the stock price model by using levy process.e.g. the Gamma process, the VG process.

Pricing European Put/Call Option with contracts on anunderlying that follows a gBm with compound jump and fSV.


Reference

Fischer Black and Myron Scholes. The Pricing of Options andCorporate Liabilities. Journal of Political Economy 81(3):637-654,1973.

D. Lamberton and B. Lapeyre. Introduction to Stochastic CalculusApplied to Finance, Chapman Hall, Great Britain at T.J. Press(Padstow) Ltd., Padstow, Cornwall, 1996.

N. H. Chan and H. Y. Wong. Simulation Techniques in FinancialRisk Management, John Wiley Sons, America, 1996.

L. Scott. Pricing Stock Options in a Jump-Diffusion Model withStochastic Volatility and Interest Rates: Applications of FourierInversion Methods, Mathematical Finance, 7(4)(1997), 413-424.

R, Merton. Option pricing when underlying stock return arediscontinuous, Journal of Financial Economics 3(1976): 125-144.


Reference (cont.)

P. Sattayatham, A. Intarasit, and A. P. Chaiyasena, A FractionalBlack-Scholes Model with Jumps, Vietnam Journal of Mathematics35:3(2007) 1-15.

J. Hull and A. White. The Pricing of Options on Assets withStochastic Volatilities, Journal of Finance, 42(1987), 281300.

M. Danilo and V. Spokoinyi. Estimation of time dependent volatilityvia local change point analysis, WIAS Preprint No. 904 (2004).

L. Goldentyer, F. C. Klebaner and R. Liptser. Tracking volatility,Problems of Information Transmition, 41(2005), 3250.

S, Heston. A closed-form solution for options with stochasticvolatility with applications to bond and currency options, Review ofFinancial Studies 6(1993): 327-343.


Reference (cont.)

J.C. Cox, J.E. Ingersoll and S.A. Ross A Theory of the TermStructure of Interest Rates, Econometrica 53(1985): 385-407.

D.B. Nelson ARCH models as diffusion approximation, Journal ofEconometrics, 45(1990), 738.

R. T. Baillie, T. Bollerslev, and H. O. Mikkelsen, Fractionallyintegrated generalized autoregressive conditional heteroskedasticity,Journal of Econometrics 74(1996), 3-30.

P. M. Robinson, The memory of stochastic volatility models, Journalof Econometrics 101(2001), 195-218.

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A Geometric Brownian Motion Model with Compound Poisson …science.sut.ac.th/mathematics/pairote/uploadfiles/Artit_cmu_DG.pdf · School of Mathematics, Institute of Science Suranaree