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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 Research Guideline Warm-Up Conceptual Research Framework
Outline
1 Research Guideline
2 Warm-UpStock Price ModelLiterature Reviews
3 Conceptual Research FrameworkModel DescriptionSimulation Examples
Outline Research Guideline Warm-Up Conceptual Research Framework
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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Outline Research Guideline Warm-Up Conceptual Research Framework
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.
Outline Research Guideline Warm-Up Conceptual Research Framework
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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)”.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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)”.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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)
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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)
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
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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 .
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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).
Outline Research Guideline Warm-Up Conceptual Research FrameworkStock Price Model Literature Reviews
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 FrameworkStock Price Model Literature Reviews
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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.
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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 )] =∞.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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 )] =∞.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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.
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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).
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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)
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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)
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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 ].
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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 ].
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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 ].
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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 ].
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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 ].
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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 ].
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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/.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
Figure 2: Log returns on the stock prices of K-BANK between July 2,2008 and June 30, 2009.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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)
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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)
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
Fig 5. Convergence of ARPE(3) and ARPE(4) with N = 25, 45, 55, 96
and 105.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
Reference
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R, Merton. Option pricing when underlying stock return arediscontinuous, Journal of Financial Economics 3(1976): 125-144.
Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
Reference (cont.)
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Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
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.
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Outline Research Guideline Warm-Up Conceptual Research FrameworkModel Description Simulation Examples
Reference (cont.)
T. Plienpanich, P. Sattayatham and T.H. Thao. Fractionalintegrated GARCH diffusion limit models, Journal of the KoreanStatistical Society, 38(2009), 231-238.
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