Markov Regime Switching of Stochastic Volatility Lévy Model on Approximation Mode
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.
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. Figure 1 displays historical value of SET index with the sample period from January 2011 to February 2013.
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 . 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.  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.  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  and Smith . Janczura and Weron  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 Section 2, 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.
Let be a two-dimensional standard Brownian motion on a filtered probability space with filtration , the -augmentation of the filtration generated by . We consider two financial assets with a risk-free asset (such as a bank account or bond) with price dynamics described by where the interest-rate process is continuous. The return process of the risky asset is described by a diffusion model to incorporate the Lévy jump; its variance is an SV process as in the following general form: where the stock-drift process and the dividend-yield process are continuous processes, and is an SV process. The notation is defined by 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 where Let be the family of Borel sets whose closure does not contain . Let denote the indicator function of the set . Let and the process be a Poisson random measure of in defined by with Lévy measure of the price process given by Here denotes the expectation with respect to measure . The jump process associated with the price process is defined, for each , via where is defined as (4). The process is called the compensated Poisson random measure.
Given a correlation process between and we introduce the Brownian motion independent of and write the usual Cholesky factorization: with correlation .
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 almost everywhere (a.s.). Under the historical measure , the asset price at time is then given by
The process 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 usual for the discounted stock price process with the bank account being the natural numéraire and get from Itô’s formula again where the Lévy jump martingale is defined by and the process is given by SDE (3). The process is the compensated Poisson random measure as defined in (6).
To determine the equivalent martingale measure under which discounted price processes are (local) -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 . For define the approximation process of the process by where , , and are continuous functions from to , which are the approximation functions of , , and , respectively. Assume in addition that the total mass of the Lévy jump measure exceeds so that By this assumption, we define where with . For , fix , , and such that a martingale condition satisfies setting so that the process defined by exists for . Define a probability measure on by Assume that Then is an equivalent local martingale measure for .
Under , let us assume that the Novikov condition holds; that is, where . The notation then denotes the expectation with respect to the measure . Then the processes are Brownian motions with respect to so that we define the random measure by where the Lévy measure is given by with a -intensity measure and function defined as above and satisfying the martingale condition (16). Then is a local martingale.
Under the risk-neutral measure , we rewrite the process of discounted stock price (11) in terms of these processes: so that the process of discounted price follows Using the martingale condition (16), the process is a local martingale under and or equivalently, Therefore the dynamic of the under the risk-neutral martingale measure is
Proposition 2. Suppose that the asset return process is governed by SED (30). Define ; then under the risk-neutral measure ,
Proof. In a similar way to that of Proposition 1, the proof follows from the Itô formula applied to . 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 where is the rate of reversion computed by the long run mean of the process and is the log rate at which tends to . The risk-neutral counterparts to these equations are, respectively, where is defined in a similar way to (9) on a martingale measure . All subscripted variables with risk-neutral measure represent risk-neutral versions of the actual variables. By Proposition 1 and definition (6) of , the SDE (33) has solution: The process is a Poisson random measure in with Lévy measure , for a measure satisfying . The last term of (37), representing the sum of the big jumps, is a compound Poisson process with intensity of jumps and jump distribution . Over finite intervals this sum is finite since any cádlág path of Lévy process has only finite number of big jumps with absolute jump size larger than . The third term of (37) is the limit of compensated Poisson processes as follows: for , 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 .
Here we use the approximation theorem for the distribution of Lévy process developed in  but modify it in accordance with our model (37). Let be given and define the following processes: where , and Define and suppose the has no atoms in a neighborhood of the origin. Introduce a candidate approximation of SDE (37) in distribution sense: If then the process converges in distribution to a standard Brownian motion , and by (43), it holds that Therefore, the distribution of (37) (with infinite jump activity) can be approximated closely to , 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 process , where satisfies the SED (36) for simulation purposes. The process is obtained using stochastic calculus transformation techniques [10, Theorem 1]. Let . Applying Itô’s lemma to the process , we obtain the following equation: Then the solution of the stochastic process is where Calibration and simulation are done in a discrete time framework. For simulation purposes using the forward Euler discretization scheme, we divide the time horizon of years into subintervals of equal length , where is a positive integer. For each , the th subinterval is represented by where . Let denote . The following proposition shows that the simulation of the volatility process is best performed by using (46).
Proposition 3. The exact value of the realization of the solution at time is generated by the following reclusive formula: where is the distributed increment of the Brownian motion on and is the length of the time discretization subinterval .
For numerical experiments, we simplified implementation to generate processes that only apply explicit schemes to (48).
The discretized version of the risk-neutral log return process is deduced from the approximation of each component as follows: let be given and let . Rewrite the for log returns: As in the Euler scheme, an integrand over , is approximated by its value at . We now approximate the first integral term using Following the first integral term, we get with . From the remaining integral in (51), we get where is the jump process of caused by the jump of , as denoted by Note that this process consists of the (possible) jumps in at arising from the jump in .
With these approximations, the discretized version of the risk-neutral log return process is displayed in the following proposition.
Proposition 4. Based on Euler approximation, the discretization scheme for is as follows: with following from the scheme (48).
Proof. We use the approximation of (52), (53), and (54) to approximate . With the transformation , the discretized version of the risk-neutral is We apply this recursively at any and replace the increment of Brownian motion with , . More explicitly, we have (56).
From (56) and setting , we have We can now calibrate the model parameters in this equation. Let us consider the parameter set for , given by . The remaining parameters are fixed that calibrate the parameter based on historical data.
The conditionals of the process are normal and independent, due to the locally constancy of the SV on small . The log-likelihood function for the asset return and is then given by The probability density function (pdf) of process given together with the volatility given has a conditional Gaussian distribution where and This pdf is the sum of the conditional probability densities weighted by the probability of the conditioning variables of the number of jumps which counts the jumps between time and . Observe that if 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 . Then
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 be a continuous-time Markov chain on finite space and the transition rate matrix where the transition rate is calculated with respect to the natural filtration generated by continuous-time Markov chain , and given by By definition, the Markov chains have the probability of a transition from state to state not depending on the global time and depending only on the time interval available for the transition; that is, for all . The regime switching (RS) of SVL model (35)-(36) is defined by the following SDE: The processes , , , , , and are constants taking values, respectively, in , , , , , and such that The solution of (65) with SV (66) is called an RS of SVGL model. Let be the filtration of historical value of the process untial time . By Proposition 4, its approximation of the solution in discrete version is given by
In the case that the future regime of the economy has only 2 states , , the conditional distribution density of , the log price of , has associated parameters across regimes given by up to the conditional probability .
Assume that a Markov switching or jump process is independent of Brownian motion . Combining both regime switching and jumps with the locally constant of the SV process , with conditional density of , from (62), we get Substituting (70) into (69) gives the conditional density of used in constructing the likelihood function for the data.
The set of the parameters is estimated by the EM algorithm (for more details see ). 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, for all . Fix the maximum number of iterations and approximation level . Apply the EM algorithm to find filtered probability with density function of (70) and smoothed probability using data generated by SV of (48) as a variance process of discretization for approximation process .(3)Generate the process .
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 fix and 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 with and and being fixed. The data cover the sample period, January 1, 2011, through February 15, 2012. In estimating the (58), we fix , somewhat arbitrarily, at .
Implementation of the Matlab program following Algorithm 5. The log prices 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 . Calibration results for model (58) with two independent regimes fitted to the SET index log prices are displayed in Figure 2.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This research is especially dedicated to the Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus. And this research project has been financially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand, under the individual project at 2011, Grant no. RS-2-54-05-1.
B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Universitext, Springer, Berlin, Germany, 2005.View at: MathSciNet
N. H. Bingham and R. Kiesel, Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives, Springer Finance, Springer, London, UK, 2nd edition, 2004.View at: MathSciNet
B. F. Hanson, “Stochastic calculus of Heston's stochastic-volatility model,” in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS '10), pp. 2423–2429, Budapest, Hungary, July 2010.View at: Google Scholar