# Markov Regime Switching of Stochastic Volatility Lévy Model on Approximation Mode

Authors: Arthit Intarasit
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper deals with financial modeling to describe the behavior of
asset returns, through consideration of economic cycles together with
the stylized empirical features of asset returns such as fat tails. We
propose that asset returns are modeled by a stochastic volatility Lévy process incorporating a regime switching model. Based on the risk-neutral
approach, there exists a large set of candidates of martingale
measures due to the driving of a stochastic volatility Lévy process in
the proposed model which renders the market incomplete in general.
We first establish an equivalent martingale measure for the proposed
model introduced in risk-neutral version. Regime switching of stochastic
volatility Lévy process is employed in an approximation mode for
model calibration and the calibration of parameters model done based
on EM algorithm. Finally, some empirical results are illustrated via
applications to the Bangkok Stock Exchange of Thailand index.


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## Body

## 1. Introduction

The market of stock products is one of the fastest growing main segments in the finance market industry today. Although the financial credit derivative industry has increased in market size, particular stock market sector investments are still attractive for all investors. We can see that in the Stock Exchange of Thailand, between 2011 to 2013, the Bangkok Stock Exchange of Thailand (SET) index, a major stock market index tracking the performance of all common stocks listed on the SET market, showed high volatility in movement behavior. The SET index average went up about 36%: from 1,025 index points in 2011 to 1,392 index points in 2012, with fluctuations in its movement in 2013. Historically, the SET averaged 727 index points from 1987 to 2013, with an all time record high of 1754 index points in January of 1994 and a record low of 207 index points in September of 1998. Figure1 displays historical value of SET index with the sample period from January 2011 to February 2013.Historical value of SET index with sample period from January 2011 to February 2013 (a) and empirical distribution of daily log returns for the SET index and fitted normal distribution (b).
(a)
(b)A pricing of SET index movement that takes into account fluctuation and high volatility has become necessary. We need models that can capture the behavior of asset prices more accurately in order to handle trade risks. Recently, continuous-time financial models have been intensively investigated in explorations to capture the stylized empirical features of asset prices or returns such as long memory, fat tails, high kurtosis, volatility clustering, and leverage. On the other hand, a new generation of financial models are able to reproduce the different phase of the business cycles and capture the cyclical behavior of the economic growth. Known as regime switching models, they were first proposed by Hamilton [1]. Succeeding Hamilton’s regime switching models, quite a few number of researchers have utilized the regime switching approach to help their models become more realistic in parameter estimation and forecasting accuracy. Thus the class of regime switching models has extended in many directions.In their development of a continuous framework, Elliott et al. [2] proposed the use of the exponential of a pure jump process with finite states of a Markov-switching compensator. On the other hand, Siu et al. [3] generalized a jump diffusion model with a Markov-switching compensator and used it to value participating life insurance products. Combining stochastic volatility models together with regime switching has been modeled for short-term interest rates, for example, Kalimipalli and Susmel [4] and Smith [5]. Janczura and Weron [6] discussed the calibration models based on the EM algorithm built on the mean-reverting process combined with Markov regime switching.It is know that Lévy processes are a class of stochastic processes that help us capture a financial asset aspect of a more realistic model such as the phenomenon of jump in asset prices or the implied volatility smile in option markets, showing that the risk-neutral returns are non-Gaussian and leptokurtic. Although this recent modeling of asset returns by jump diffusion allows for regime switching, few studies have explored models of diffusion with Lévy jump incorporating stochastic volatility and switching in regimes for modeling an asset return. Motivated by this fact, we propose a jump diffusion process including its variance as a stochastic volatility and the asset return considered on Markov’s regime switching models to facilitate the matching of the empirical distribution with the asset return founded in real economic data. Under the risk-neutral approach, we construct and study our proposed model in a risk-neutral world. Approximation of proposed model with special structures is presented to avoid complexity of numerical computation and to suggest a suitable consistent approximation model of the proposed model.The rest of paper is organized as follows. In Section2, we introduce the dynamic of asset return described by the stochastic volatility Lévy process. In fact, the model is a bivariate-stochastic differential equation, whose solution of proposed dynamics is obtained by using the Itô formula for Lévy processes. Using the approach of martingale modeling, we construct a martingale measure and introduce the risk-neutral model of proposed model in Section 3. In Section 4 model specification for implementation is introduced and its approximation is considered in Section 5. We state our proposed model incorporating regime switching in Section 6. Finally, we present some empirical results from applying our model to real market financial data.


## 2. Modeling Asset Price Dynamics

In order to model financial asset prices in the market, we introduce Lévy jump diffusion (LJD) with stochastic volatility (SV), a bivariate-stochastic differential equation (SDE) type, as follows.LetW=(W1,W2) be a two-dimensional standard Brownian motion on a filtered probability space (Ω,ℱ,P,F) with filtration F=(ℱt)0≤t≤T, the P-augmentation of the filtration generated by W. We consider two financial assets (B,S) with a risk-free asset (such as a bank account or bond) with price dynamics described by
(1)dBt=rtBtdt,B0=1,
where the interest-rate process r:ℜ→ℜ is continuous. The return process of the risky asset S is described by a diffusion model to incorporate the Lévy jump; its variance is an SV process as in the following general form:
(2)dSt=(μt-qt)St-dt+σ(vt)St-dW1,t+St-dZt,(3)dvt=a(t,St,vt)dt+b(t,St,vt)dW2,t,
where the stock-drift process μ:[0,T]→ℜ and the dividend-yield process q:[0,T]→ℜ are continuous processes, and σ(·) is an SV process. The notation St- is defined by
(4)St-∶=lims↑tSs,
which means that whenever there is a jump, the value of the process before the jump is used on the left-hand side of (2). The last term of (2), the differential form of the jump derived by the Lévy process, is defined in the form of Lévy-Itô decomposition as
(5)Zt∶=∫0t‍∫ℜ‍zN¯(ds,dz),
where
(6)N¯(dt,dz)={N(dt,dz)-ν(dz)dt=:N~if|z|<1,N(dt,dz)if|z|≥1.
Let B0 be the family of Borel sets A⊂ℜ whose closure A¯ does not contain 0. Let χA denote the indicator function of the set A. Let A∈B0 and the process N be a Poisson random measure of S(·) in ℜ+×ℜ0(ℜ0∶=ℜ∖{0}) defined by
(7)N((s,t]×A)∶=∑u∈(s,t]:ΔS≠0‍χA(ΔSu)
with Lévy measure ν of the price process S given by
(8)ν(A)∶=E[N(1,A)]=E[∑u∈(s,t]:ΔS≠0‍χA(ΔSu(ω))].
Here E=EP denotes the expectation with respect to measure P. The jump process ΔS∶=(ΔSt)0≤t≤T associated with the price process S is defined, for each 0≤t≤T, via ΔSt∶=St-St- where St- is defined as (4). The process N~ is called the compensated Poisson random measure.Given a correlation processρ between dW1,t and dW2,t we introduce the Brownian motion dW~2,t independent of dW1,t and write the usual Cholesky factorization:
(9)dW2,t=ρdW1,t+1-ρ2dW~2,t
with correlation ρ∈(-1,1).Furthermore we assume that all processes are bounded and sufficiently smooth to guarantee unique strong solutions of the various stochastic differential equations that we encounter.The following proposition provides an explicit solution of SDE (2).Proposition 1.
Suppose that some risky assets have a dynamics of return given by SDE (2) with an initial value S0=S(0) almost everywhere (a.s.). Under the historical measure P, the asset price St at time t is then given by
(10)St=S(0)exp{∫0t‍(μs-qs-12σ2(vs))ds+∫0t‍σ(vs)dW1,s+∫0t‍∫|z|<1‍{ln(1+z)-z}ν(dz)ds+∫0t‍∫ℜ‍ln(1+z)N¯(ds,dz)}.Proof.
If we define the processXt=lnSt, such that St corresponds to SDE (2), then it follows from the Itô’s formula (see [7, Theorem  1.14]) with S0=S(0) a.s. that the process (10) is the explicit solution of SDE (2).The processSt of (10) is called the stochastic volatility of geometric Lévy model (SVGL) which is used to model the process of risky asset pricing in this paper.


## 3. Risk-Neutral Dynamics of Asset Price

Incorporating a Lévy jump and/or stochastic volatility in a diffusion model of asset returns leads to a market being incomplete. As a result there are different choices of equivalent martingale measure. By risk-neutral modeling, we should determine the dynamic price of asset return in the risk-neutral version and choose a pricing measure form various equivalent martingale measures.We write as usualS~t=St/Bt for the discounted stock price process with the bank account being the natural numéraire and get from Itô’s formula again
(11)dS~t=(μt-qt-rt)S~t-dt+σ(vt)S~t-dW1,t+S~t-dZ~t,
where the Lévy jump martingale Z~t is defined by
(12)Z~t=∫0t‍∫ℜ‍zN~(ds,dz),
and the process vt is given by SDE (3). The process N~ is the compensated Poisson random measure as defined in (6).To determine the equivalent martingale measureQ under which discounted price processes S~t are (local) ℱt-martingales, we rely on Girsanov’s theorem for semimartingales (see, e.g., [8, Theorem  7.4.1]). In our model, this follows from the special structure of the Girsanov density used to perform a change of measure in setting of both Brownian motion and Lévy jump. Define the market price process of risk θ1,t∶=μt-qt-rt. For ε>0 define the approximation process of the process θ1,t by
(13)θε,t∶=με,t-qε,t-rε,tσ(vt),
where με, qε, and rε are continuous functions from [0,T] to ℜ, which are the approximation functions of μ, q, and r, respectively. Assume in addition that the total mass of the Lévy jump measure ν exceeds θ1,t-θε,t so that
(14)∫ℜ‍ν(dz)>θ1,t-σ(vt)θε,tz.
By this assumption, we define
(15)θ3,z∶=(θ1,t-σ(vt)θε,t)zν(A)χA(z),
where A⊂ℜ with θ1,t-σ(vt)θε,t<ν(A)<∞. For ε>0, fix θε,t,  θ1,t, and θ3,z<1 such that a martingale condition satisfies
(16)θ1,t-θε,tσ(vt)-∫ℜ‍zθ3,zν(dz)=0,
setting
(17)θ2,t=-11-ρ2(ρθε,t+θ1,tb(t,St,vt)),
so that the process Lt defined by
(18)logLt=-∫0t‍θ1,sdW1,s-∫0t‍θ2,sdW2,s-12∫0t‍(θ1,s2+θ2,s2)ds+∫0t‍∫ℜ‍ln(1-θ3,z)N~(ds,dz)+∫0t‍∫ℜ‍[ln(1-θ3,z)+θ3,z]ν(dz)ds
exists for 0≤t≤T. Define a probability measure Q on ℱT by
(19)dQ(ω)=L(T)dP(ω).
Assume that
(20)EP[L(T)]=1.
Then Q is an equivalent local martingale measure for S~t.UnderQ, let us assume that the Novikov condition holds; that is,
(21)EP[exp(12∫0T‍θi,s2ds)]<∞,
where i=1,2. The notation EP then denotes the expectation with respect to the measure P. Then the processes
(22)W1,tQ=W1,t+∫0t‍θε,sds,W~2,tQ=W~2,t+∫0t‍θ2,sds
are Brownian motions with respect to Q so that we define the random measure N~Q(·,·) by
(23)N~Q(dt,dz)∶=N(dt,dz)-νQ(dt,dz),
where the Lévy measure νQ is given by
(24)νQ(dt,dz)=(1-θ3,z)ν(dz)dt,
with a P-intensity measure ν(dz)dt and function θ3,z defined as above and satisfying the martingale condition (16). Then
(25)∫0t‍∫ℜ‍N~Q(ds,dz)=∫0t‍∫ℜ‍N(ds,dz)-∫0t‍∫ℜ‍(1-θ3,z)ν(dz)ds
is a Q local martingale.Under the risk-neutral measureQ, we rewrite the process of discounted stock price (11) in terms of these processes:
(26)dS~tS~t-=(μt-qt-rt)dt+σ(vt)dW1,t+∫ℜ‍zN~(dt,dz)=(μt-qt-rt-θε,tσ(vt))dt+σ(vt)dW1,tQ+∫ℜ‍zN(ds,dz)-∫ℜ‍zν(dz)dt=(θ1,t-θε,tσ(vt))dt+σ(vt)dW1,tQ-∫ℜ‍zν(dz)dt+∫ℜ‍z{N~Q(ds,dz)+(1-θ3,s(z))ν(dz)dt}
so that the process of discounted price follows
(27)dS~t=S~t-{(θ1,t-θε,tσ(vt)-∫ℜ‍zθ3,sν(dz))dt+σ(vt)dW1,tQ+∫ℜ‍zN~Q(ds,dz)(θ1,t-θε,tσ(vt)-∫ℜ‍zθ3,sν(dz))}.
Using the martingale condition (16), the process S~t is a local martingale under Q and
(28)dS~t=S~t-(σ(vt)dW1,tQ+∫ℜ‍zN~Q(ds,dz)),
or equivalently,
(29)dSt=St-(rtdt+σ(vt)dW1,tQ+∫ℜ‍zN~Q(ds,dz)).
Therefore the dynamic of the (S,v) under the risk-neutral martingale measure Q is
(30)dSt=St-(rtdt+σ(vt)dW1,tQ+∫ℜ‍N~Q(ds,dz))(31)dvt=(a(t,St,vt)+θ1,t)dt+b(t,St,vt)(ρtdW1,tQ+1-ρt2dW~2,tQ).Proposition 2.
Suppose that the asset return processSt is governed by SED (30). Define X~t=lnSt; then under the risk-neutral measure Q,
(32)dX~t=∫0t‍(rs-12σ2(vs))dt+∫|z|<1‍{ln(1+z)-z}νQ(dt,dz)+∫0t‍σ(vs)dW1,sQ+∫ℜ‍ln(1+z)N~Q(dt,dz).Proof.
In a similar way to that of Proposition1, the proof follows from the Itô formula applied to X~t=ln(St). We omit the details.


## 4. Model Specification for Implementation

Both of the SDEs (2) or (30) together with (3) or (31), respectively, are expressed in an abstract form in a very general setting enabling many different processes of asset price to be used. Here, we will use a jump diffusion process with variance following a square root of a stochastic process as an example. The full model is
(33)dSt=St-((μ-q)dt+vtdW1,t+∫ℜ‍zN¯(ds,dz)),(34)dvt=β(m-vt)dt+σvtdW2,t,
where m is the rate of reversion computed by the long run mean of the process v∶=(vt)t≥0 and β is the log rate at which v tends to m. The risk-neutral counterparts to these equations are, respectively,
(35)dSt=St-(rdt+vtdW1,tQ+∫ℜ‍zN~Q(ds,dz)),(36)dvt=((μ-q-r)+β(m-vt))dt+σvtdW2,tQ,
where dW2,tQ is defined in a similar way to (9) on a martingale measure Q. All subscripted variables with risk-neutral measure Q represent risk-neutral versions of the actual variables. By Proposition 1 and definition (6) of N¯, the SDE (33) has solution:
(37)St=S(0)exp{∫0t‍(μ-q-12vs)ds+∫0t‍vsdW1,s+∫0t‍∫|z|<1‍{ln(1+z)-z}ν(dz)ds+∫0t‍∫|z|<1‍ln(1+z)N~(ds,dz)+∫0t‍∫|z|≥1‍ln(1+z)N(ds,dz)}.
The process N is a Poisson random measure in ℜ+×ℜ0 with Lévy measure ν(dt,dx)=ν(dx)dt, for a measure ν satisfying ∫ℜ0‍min(1,z2)ν(dz)<∞. The last term of (37),
(38)∫0t‍∫|z|≥1‍ln(1+z)N(ds,dz),
representing the sum of the big jumps, is a compound Poisson process with intensity of jumps ν(|z|>1) and jump distribution χ(|z|≥1)ν(dx)/ν(|z|≥1). Over finite intervals [0,t] this sum is finite since any cádlág path of Lévy process S has only finite number of big jumps with absolute jump size larger than 1. The third term of (37) is the limit of compensated Poisson processes as follows: for ε>0,
(39)∫0t‍∫|z|<1‍ln(1+z)N~(ds,dz)=limε↓0∫0t‍∫ε<|z|<1‍ln(1+z)N~(ds,dz)
represents the sum of the small jumps. In general there are too many jumps to get convergence but this sum can converge by compensating. Moreover, the sum of the small jumps of the solution of SDE (35) converges due to the jump part of SED (35) driven by the compensated risk-neutral measure Q.Here we use the approximation theorem for the distribution of Lévy process developed in [9] but modify it in accordance with our model (37). Let ε>0 be given and define the following processes:
(40)St,ε‡=S(0)exp{∫0t‍bs,εds+∫0t‍vsdW1,s+∫0t‍∫|z|≥ε‍ln(1+z)N(ds,dz)},
where bs,ε∶=μ-q-(1/2)(vs)-∫ε<|z|≤1‍{ln(1+z)-z}ν(dz), and
(41)St,ε∶=St-St,ε‡=∫0t‍∫|z|<1‍{ln(1+z)-z}ν(dz)ds+∫0t‍∫|z|<ε‍ln(1+z)N~(ds,dz).
Define σε2∶=∫|z|≤ε‍z2ν(dz) and suppose the ν has no atoms in a neighborhood of the origin. Introduce a candidate approximation of SDE (37) in distribution sense:
(42)St,ε†=S(0)exp{∫0t‍bs,εds+∫0t‍vs2+σε2dW1,s+∫0t‍∫|z|≥ε‍ln(1+z)N(ds,dz)}.
If
(43)limε→01εσε=∞,
then the process (vε-1St,ε)t≥0 converges in distribution to a standard Brownian motion (Bt)t≥0, and by (43), it holds that
(44)supx∈ℜ|P(St≤x)-P(St,ε†≤x)|≤cεσε.
Therefore, the distribution of St (37) (with infinite jump activity) can be approximated closely to St,ε†, the combination of a Wiener process with drift and a compound Poisson process.An approximation to the solution of SED (35) may be obtained in the same way.


## 5. Approximation of Asset Price Dynamics

We begin by finding an explicit formula of the stochastic processv=(vt)0≤t≤T, where vt satisfies the SED (36) for simulation purposes. The process vt is obtained using stochastic calculus transformation techniques [10, Theorem  1]. Let Vt=F(t,vt)∈C1,2(ℜ). Applying Itô’s lemma to the process V, we obtain the following equation:
(45)dVt=Ft(t,vt)dt+Fv(t,vt)dvt+12Fvv(t,vt)σv2vtdt.
Then the solution of the stochastic process v is
(46)vt=e-β(Vt2)2,
where
(47)Vt=2v0+2Ig(t),Ig(t)=12∫0t‍eβ/2((βm-σ2/4vs)ds+σdW2,sQ).
Calibration and simulation are done in a discrete time framework. For simulation purposes using the forward Euler discretization scheme, we divide the time horizon [0,T] of T years into N0T subintervals of equal length Δ∶=T/N0, where T is a positive integer. For each n=1,…,N0, the nth subinterval is represented by [tn,tn-1] where tn=nΔ. Let vn denote vtn. The following proposition shows that the simulation of the volatility process v is best performed by using (46).Proposition 3.
The exact value of the realization of the solutionv at time tn+1 is generated by the following reclusive formula:
(48)vn+1=v0+2vnI~g(tn)+I~g2(tn),
where
(49)I~g(tn)=12eβ/2((βm-σ2/4)Δnvn+ΔW2,nQ),ΔW2,nQ=∫tntn+1‍dW2,tQ=W2,tn+1Q-dW2,tnQ
is the 𝒩(0,Δn) distributed increment of the Brownian motion W on [tn,tn+1] and
(50)Δn=∫tntn+1‍dt=tn+1-tn
is the length of the time discretization subinterval [tn,tn+1].For numerical experiments, we simplified implementation to generate processesv that only apply explicit schemes to (48).The discretized version of the risk-neutral log return processSt,ε† is deduced from the approximation of each component as follows: let Δ>0 be given and let Xt†∶=logSt,ε†. Rewrite the St,ε† for log returns:
(51)Xt+Δ†=Xt†+∫tt+Δ‍bs,εds+∫tt+Δ‍vs2+vε2dW1,s+∫tt+Δ‍∫|z|≥ε‍ln(1+z)N(ds,dz).
As in the Euler scheme, an integrand over [t,t+Δ], is approximated by its value at t. We now approximate the first integral term using
(52)∫tt+Δ‍bs,εds≈bt,εΔ.
Following the first integral term, we get
(53)∫tt+Δ‍vs2+vε2dW1,s≈(vt2+vε2)ΔW1,t,
with ΔW1,t∶=W1,t+Δ-W1,t. From the remaining integral in (51), we get
(54)∫tt+Δ‍∫|z|≥ε‍ln(1+z)ΔN(ds,dz)≈(ΔNSt,ε†)Δ,
where ΔNSt,ε† is the jump process of St,ε† caused by the jump of N(t,z), as denoted by
(55)ΔNSt,ε†=∫|z|≥ε‍ln(1+z)N(t,dz).
Note that this process consists of the (possible) jumps in X at t arising from the jump in N(t,·).With these approximations, the discretized version of the risk-neutral log return processXt is displayed in the following proposition.Proposition 4.
Based on Euler approximation, the discretization scheme forSt,ε† is as follows:
(56)Sn+1,ε†=Sn,ε†exp{bn,εΔn+vn2+vε2(ΔnZn+1)+(ΔNSn,ε†)Δnvn2+vε2}
with vn following from the scheme (48).Proof.
We use the approximation of (52), (53), and (54) to approximate St+Δ,ε†. With the transformation S†=exp(X†), the discretized version of the risk-neutral St+Δ,ε† is
(57)St+Δ,ε†≈St,ε†exp{bt,εΔ+vt2+vε2ΔW1,t+(ΔNSt,ε†)Δ}.
We apply this recursively at any Δ=Δn∶=tn+1-tn>0 and replace the increment of Brownian motion W with ΔZ,  Z~𝒩(0,1). More explicitly, we have (56).From (56) and setting Xt,ε†∶=logSt+Δ,ε†, we have
(58)Xt+Δ,ε†≈Xt†+bt,εΔ+vt2+vε2ΔW1,t+(ΔNSt,ε†)Δ.
We can now calibrate the model parameters in this equation. Let us consider the parameter set for X†, given by Θ=(r,μ,β,m,σ,λ,σY). The remaining parameters q are fixed that calibrate the parameter based on historical data.The conditionals of the processX† are normal and independent, due to the locally constancy of the SV v on small Δt. The log-likelihood function for the asset return Xn† and vn is then given by
(59)ℒ(Θ)=∑n=1N0‍logf(Xn†,vn;Θ).
The probability density function (pdf) of process Xn given Xn-1 together with the volatility vn given vn-1 has a conditional Gaussian distribution
(60)f(xn+1†,vn+1∣Θ)=∑j=0+∞ℚ(nt=j)×f𝒩(xn+1†,vn+1;bn,εΔn+xn†+jμY,v^n2Δn+jσY2),
where v^n:=vn2+vε2 and
(61)f𝒩(xn+1†,vn+1;Θ)=12π(v^n2+jσY2)exp(-(xn+1†-bn,ε-xn†-jμY)22(v^n2+jσY2)).
This pdf is the sum of the conditional probability densities weighted by the probability of the conditioning variables of the number of jumps nt=Nt-Nt-Δt which counts the jumps between time t-Δt and t. Observe that if Δt is small, typically the Poisson process jumps at most one as a result of a mixture of two Gaussian random variables weighted by the probability of zero or one jump in Δt. Then
(62)f(xn+1†;Θ)=(1-λtn+1)f𝒩(xn+1†,vn+1;bn,εΔn+x†,v^n2Δn)+λtn+1f𝒩(xn+1†,vn+1;bn,εΔn+xn†+μY,v^n2Δn+σY2).


## 6. The Switching-Regime Model of Asset Price Dynamics

We described our proposed model whose parameter values depend on the value of a continuous-time Markov chain.Let(αt)t∈[0,T] be a continuous-time Markov chain on finite space 𝒮∶={1,2,…,K} and the transition rate matrix Q=[qij] where the transition rate qij is calculated with respect to the natural filtration generated by continuous-time Markov chain α, ℱtα∶={σ(αs):0≤s≤t} and given by
(63)qij≥0ifi≠j∀i,j∈𝒮,qii=-∑j=0,j≠iK‍qijotherwise.
By definition, the Markov chains αt have the probability of a transition from state i to state j not depending on the global time and depending only on the time interval available for the transition; that is,
(64)ℙ(αt+s=j∣αs=i)=qij=ℙ(αt=j∣α0=i)
for all s,t≥0. The regime switching (RS) of SVL model (35)-(36) is defined by the following SDE:
(65)dSt=St-(r(αt)dt+vtdW1,tQ+∫ℜ‍zN~Q(ds,dz)),(66)dvt=(μ(αt)-q(αt)-r(αt)+β(αt)(m(αt)-vt))dt+σ(αt)vtdW2,tQ.
The processes r(αt), μ(αt), q(αt), β(αt), m(αt), and σ(αt) are constants taking values, respectively, in r(𝒮), μ(𝒮), q(𝒮), β(𝒮), m(𝒮), and σ(𝒮) such that
(67)r(𝒮)∶={r1,…,rK},μ(𝒮)∶={μ1,…,μK},q(𝒮)∶={q1,…,qK},β(𝒮)∶={β1,…,βK},m(𝒮)∶={m1,…,mK},σ(𝒮)∶={σ1,…,σK}.
The solution St(αt) of (65) with SV (66) is called an RS of SVGL model. Let ℱnS be the filtration of historical value of the process Sn untial time tn. By Proposition 4, its approximation of the solution St(αt) in discrete version is given by
(68)Sn+1,ε†=Sn,ε†exp{bn,ε(αn)Δn+vn2(αn)+vε2(ΔnZn+1)+(ΔNSn,ε†)Δnvn2(αn)+vε2}.The regime switching version of model (33) with (34) and its solution are defined by in similar manner.In the case that the future regime of the economy has only 2 statesαk=k,  k=1,2, the conditional distribution density of Xn+1,ε†, the log price of Sn+1,ε†, has associated parameters across regimes given by
(69)f(Xn+1,ε†∣ℱnα;Θ)=f(Xn+1,ε†∣ℱnα,αn=1;Θ1)q1,n+f(Xn+1,ε†∣ℱn-1α,αn=2;Θ2)(1-q1,n)
up to the conditional probability q1,n=Q{αn=1∣ℱnα}.Assume that a Markov switching or jump process is independent of Brownian motionW1Q,W2Q. Combining both regime switching and jumps with the locally constant of the SV process v, with conditional density of Xn,ε†, from (62), we get
(70)f(Xn+1,ε†∣ℱn+1X†,αn=k;Θk)=(1-λtn+1)f(Sn,ε†∣ℱnX†,αn=j,nojump;Θk)+λtn+1f(Xn,ε†∣ℱn-1X†,αn=i,jump;Θk)=(1-λtn+1)f𝒩(xn†,vn;bn,εΔn+xn,v^n2Δn)+λtn+1f𝒩(xn†,vn;bn,εΔn+xn+jμY,v^n2Δn+σY2).
Substituting (70) into (69) gives the conditional density of Xn,ε† used in constructing the likelihood function for the data.The set of the parametersΘ is estimated by the EM algorithm (for more details see [6]). The procedure for implementation to real historical data is as follows.Algorithm 5.
Consider the following.(1)
Estimate initial parameters of regime processes from historical data using a numerical scheme of model (58) with (48) to maximize the likelihood function (62) through a solver.(2)
Set an initial vector,Θ(0)∶=(rk0,σk0,mk0,βk0,μY,k0,σY,k0) for all k∈𝒮. Fix the maximum number of iterations N and approximation level ε>0. Apply the EM algorithm to find filtered probability with density function of (70) and smoothed probability using data generated by SV vn of (48) as a variance process of discretization for approximation process Xn,ε.(3)
Generate the processXn,ε.


## 7. Application Example of Thailand SET Index

In this section we provide numerical results from the proposed model. We consider a stock index for the Bangkok Stock Exchange of Thailand (SET) index from January 1, 2011, to February 2013. We fixr and q at 0.025 and 0.125, respectively. The estimates for the proposed model (58) with SV (48) using daily data on continuously compounded SET index returns are σ=0.7500,m=0.055,β=0.4500 with μY=0.2250 and σY=0.1250 and being fixed. The data cover the sample period, January 1, 2011, through February 15, 2012. In estimating the (58), we fix ε, somewhat arbitrarily, at 0.5.Implementation of the Matlab program following Algorithm5. The log prices Xt,ε† in the case of proposed model (58) are the base regime and the second regime representing the jump prices is given by i.i.d. log-normal distribution log(Xt,α2)~𝒩(0.3250,0.485). Calibration results for model (58) with two independent regimes fitted to the SET index log prices are displayed in Figure 2.Calibration results for model (56) with two independent regimes fitted to the SET index log prices. The prices jump is denoted by dots in the upper panel and the lower panel shows the conditional probability of beginning of the jump regime.
(a)
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*Source: 549304-2013-12-04.xml*