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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

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 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 [

In their development of a continuous framework, Elliott et al. [

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

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

Given a correlation process

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 (

Suppose that some risky assets have a dynamics of return given by SDE (

If we define the process

The process

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

To determine the equivalent martingale measure

Under

Under the risk-neutral measure

Suppose that the asset return process

In a similar way to that of Proposition

Both of the SDEs (

Here we use the approximation theorem for the distribution of Lévy process developed in [

An approximation to the solution of SED (

We begin by finding an explicit formula of the stochastic process

The exact value of the realization of the solution

For numerical experiments, we simplified implementation to generate processes

The discretized version of the risk-neutral log return process

With these approximations, the discretized version of the risk-neutral log return process

Based on Euler approximation, the discretization scheme for

We use the approximation of (

From (

The conditionals of the process

We described our proposed model whose parameter values depend on the value of a continuous-time Markov chain.

Let

The regime switching version of model (

In the case that the future regime of the economy has only 2 states

Assume that a Markov switching or jump process is independent of Brownian motion

The set of the parameters

Consider the following.

Estimate initial parameters of regime processes from historical data using a numerical scheme of model (

Set an initial vector,

Generate the process

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

Implementation of the Matlab program following Algorithm

Calibration results for model (

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.