Markov Regime Switching of Stochastic Volatility Lévy Model on Approximation Mode

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.

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. Figure 1 displays historical value of SET index with the sample period from January 2011 to February 2013.

(a)

(b)

Figure 1 

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 [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 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 𝑊=(𝑊₁,𝑊₂) be a two-dimensional standard Brownian motion on a filtered probability space (Ω,ℱ,𝑃,𝐹) with filtration 𝐹=(ℱ_𝑡)_{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 𝑑𝐵_𝑡=𝑟_𝑡𝐵_𝑡𝑑𝑡,𝐵₀=1, (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: 𝑑𝑆_𝑡=𝜇_𝑡−𝑞_𝑡𝑆_𝑡−𝑑𝑡+𝜎𝑣_𝑡𝑆_𝑡−𝑑𝑊_1,𝑡+𝑆_𝑡−𝑑𝑍_𝑡,𝑑𝑣_𝑡=𝑎𝑡,𝑆_𝑡,𝑣_𝑡𝑑𝑡+𝑏𝑡,𝑆_𝑡,𝑣_𝑡𝑑𝑊_2,𝑡, (2)(3) 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 𝑆_𝑡− is defined by 𝑆_𝑡−∶=lim_𝑠↑𝑡𝑆_𝑠, (4) 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 𝑍_𝑡∶=^𝑡_0ℜ𝑧𝑁(𝑑𝑠,𝑑𝑧), (5) where 𝑁()=𝑁()()|𝑧|𝑁()|𝑧|𝑑𝑡,𝑑𝑧𝑑𝑡,𝑑𝑧−𝜈𝑑𝑧𝑑𝑡=∶𝑁if<1,𝑑𝑡,𝑑𝑧if≥1. (6) Let 𝐁₀ be the family of Borel sets 𝐴⊂ℜ whose closure 𝐴 does not contain 0. Let 𝜒_𝐴 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 𝑁((𝑠,𝑡]×𝐴)∶=_{𝑢∈(𝑠,𝑡]∶Δ𝑆≠0}𝜒_𝐴Δ𝑆_𝑢 (7) with Lévy measure 𝜈 of the price process 𝑆 given by 𝜈(𝐴)∶=𝐸[𝑁(1,𝐴)]=𝐸_{𝑢∈(𝑠,𝑡]∶Δ𝑆≠0}𝜒_𝐴Δ𝑆_𝑢(𝜔). (8) Here 𝐸=𝐸_𝑃 denotes the expectation with respect to measure 𝑃. The jump process Δ𝑆∶=(Δ𝑆_𝑡)_{0≤𝑡≤𝑇} associated with the price process 𝑆 is defined, for each 0≤𝑡≤𝑇, via Δ𝑆_𝑡∶=𝑆_𝑡−𝑆_𝑡− where 𝑆_𝑡− is defined as (4). The process 𝑁 is called the compensated Poisson random measure.

Given a correlation process 𝜌 between 𝑑𝑊_1,𝑡 and 𝑑𝑊_2,𝑡 we introduce the Brownian motion 𝑑𝑊_2,𝑡 independent of 𝑑𝑊_1,𝑡 and write the usual Cholesky factorization: 𝑑𝑊_2,𝑡=𝜌𝑑𝑊_1,𝑡+1−𝜌²𝑑𝑊_2,𝑡 (9) 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 𝑆_𝑡 at time 𝑡 is then given by 𝑆_𝑡=𝑆(0)exp^𝑡₀𝜇_𝑠−𝑞_𝑠−12𝜎²𝑣_𝑠𝑑𝑠+^𝑡₀𝜎𝑣_𝑠𝑑𝑊_1,𝑠+^𝑡_0|𝑧|<1{()}𝜈()+ln1+𝑧−𝑧𝑑𝑧𝑑𝑠_𝑡0ℜln(1+𝑧)𝑁(𝑑𝑠,𝑑𝑧). (10)

Proof. If we define the process 𝑋_𝑡=ln𝑆_𝑡, such that 𝑆_𝑡 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 process 𝑆_𝑡 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 𝑆_𝑡=𝑆_𝑡/𝐵_𝑡 for the discounted stock price process with the bank account being the natural numéraire and get from Itô’s formula again 𝑑𝑆_𝑡=𝜇_𝑡−𝑞_𝑡−𝑟_𝑡𝑆_𝑡−𝑑𝑡+𝜎𝑣_𝑡𝑆_𝑡−𝑑𝑊_1,𝑡+𝑆_𝑡−𝑑𝑍_𝑡, (11) where the Lévy jump martingale 𝑍_𝑡 is defined by 𝑍_𝑡=^𝑡_0ℜ𝑧𝑁(𝑑𝑠,𝑑𝑧), (12) and the process 𝑣_𝑡 is given by SDE (3). The process 𝑁 is the compensated Poisson random measure as defined in (6).

To determine the equivalent martingale measure 𝑄 under which discounted price processes 𝑆_𝑡 are (local) ℱ_𝑡-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 𝜃_1,𝑡∶=𝜇_𝑡−𝑞_𝑡−𝑟_𝑡. For 𝜀>0 define the approximation process of the process 𝜃_1,𝑡 by 𝜃_𝜀,𝑡𝜇∶=^𝜀,𝑡−𝑞_𝜀,𝑡−𝑟_𝜀,𝑡𝜎𝑣_𝑡, (13) where 𝜇_𝜀, 𝑞_𝜀, and 𝑟_𝜀 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 𝜃_1,𝑡−𝜃_𝜀,𝑡 so that _ℜ𝜈()>𝜃𝑑𝑧^1,𝑡−𝜎𝑣_𝑡𝜃_𝜀,𝑡𝑧. (14) By this assumption, we define 𝜃_3,𝑧𝜃∶=^1,𝑡−𝜎𝑣_𝑡𝜃_𝜀,𝑡(𝐴)𝜒𝑧𝜈^𝐴(𝑧), (15) where 𝐴⊂ℜ with 𝜃_1,𝑡−𝜎(𝑣_𝑡)𝜃_𝜀,𝑡<𝜈(𝐴)<∞. For 𝜀>0, fix 𝜃_𝜀,𝑡,  𝜃_1,𝑡, and 𝜃_3,𝑧<1 such that a martingale condition satisfies 𝜃_1,𝑡−𝜃_𝜀,𝑡𝜎𝑣_𝑡−_ℜ𝑧𝜃_3,𝑧𝜈(𝑑𝑧)=0, (16) setting 𝜃_2,𝑡1=−1−𝜌²𝜌𝜃_𝜀,𝑡+𝜃_1,𝑡𝑏𝑡,𝑆_𝑡,𝑣_𝑡, (17) so that the process 𝐿_𝑡 defined by log𝐿_𝑡=−^𝑡₀𝜃_1,𝑠𝑑𝑊_1,𝑠−^𝑡₀𝜃_2,𝑠𝑑𝑊_2,𝑠−12^𝑡₀𝜃²_1,𝑠+𝜃²_2,𝑠+𝑑𝑠_𝑡0ℜln1−𝜃_3,𝑧𝑁()+𝑑𝑠,𝑑𝑧_𝑡0ℜln1−𝜃_3,𝑧+𝜃_3,𝑧𝜈(𝑑𝑧)𝑑𝑠 (18) exists for 0≤𝑡≤𝑇. Define a probability measure 𝑄 on ℱ_𝑇 by 𝑑𝑄(𝜔)=𝐿(𝑇)𝑑𝑃(𝜔). (19) Assume that 𝐸_𝑃[𝐿(𝑇)]=1. (20) Then 𝑄 is an equivalent local martingale measure for 𝑆_𝑡.

Under 𝑄, let us assume that the Novikov condition holds; that is, 𝐸_𝑃1exp2^𝑇₀𝜃²_𝑖,𝑠𝑑𝑠<∞, (21) where 𝑖=1,2. The notation 𝐸_𝑃 then denotes the expectation with respect to the measure 𝑃. Then the processes 𝑊^𝑄_1,𝑡=𝑊_1,𝑡+^𝑡₀𝜃_𝜀,𝑠𝑊𝑑𝑠,_𝑄2,𝑡=𝑊_2,𝑡+^𝑡₀𝜃_2,𝑠𝑑𝑠 (22) are Brownian motions with respect to 𝑄 so that we define the random measure 𝑁_𝑄(⋅,⋅) by 𝑁_𝑄(𝑑𝑡,𝑑𝑧)∶=𝑁(𝑑𝑡,𝑑𝑧)−𝜈_𝑄(𝑑𝑡,𝑑𝑧), (23) where the Lévy measure 𝜈_𝑄 is given by 𝜈_𝑄(𝑑𝑡,𝑑𝑧)=1−𝜃_3,𝑧𝜈(𝑑𝑧)𝑑𝑡, (24) with a 𝑃-intensity measure 𝜈(𝑑𝑧)𝑑𝑡 and function 𝜃_3,𝑧 defined as above and satisfying the martingale condition (16). Then ^𝑡_0ℜ𝑁_𝑄(𝑑𝑠,𝑑𝑧)=^𝑡_0ℜ𝑁()−𝑑𝑠,𝑑𝑧_𝑡0ℜ1−𝜃_3,𝑧𝜈(𝑑𝑧)𝑑𝑠 (25) is a 𝑄 local martingale.

Under the risk-neutral measure 𝑄, we rewrite the process of discounted stock price (11) in terms of these processes: 𝑑𝑆_𝑡𝑆_𝑡−=𝜇_𝑡−𝑞_𝑡−𝑟_𝑡𝑑𝑡+𝜎𝑣_𝑡𝑑𝑊_1,𝑡+_ℜ𝑧𝑁()=𝜇𝑑𝑡,𝑑𝑧_𝑡−𝑞_𝑡−𝑟_𝑡−𝜃_𝜀,𝑡𝜎𝑣_𝑡𝑑𝑡+𝜎𝑣_𝑡𝑑𝑊^𝑄_1,𝑡+_ℜ𝑧𝑁(𝑑𝑠,𝑑𝑧)−_ℜ()=𝜃𝑧𝜈𝑑𝑧𝑑𝑡_1,𝑡−𝜃_𝜀,𝑡𝜎𝑣_𝑡𝑑𝑡+𝜎𝑣_𝑡𝑑𝑊^𝑄_1,𝑡−_ℜ𝑧𝜈(𝑑𝑧)𝑑𝑡+_ℜ𝑧𝑁_𝑄(𝑑𝑠,𝑑𝑧)+1−𝜃_3,𝑠(𝑧)𝜈(𝑑𝑧)𝑑𝑡 (26) so that the process of discounted price follows 𝑑𝑆_𝑡=𝑆_𝑡−𝜃_1,𝑡−𝜃_𝜀,𝑡𝜎𝑣_𝑡−_ℜ𝑧𝜃_3,𝑠𝜈()𝑣𝑑𝑧𝑑𝑡+𝜎_𝑡𝑑𝑊^𝑄_1,𝑡+_ℜ𝑧𝑁_𝑄(𝑑𝑠,𝑑𝑧). (27) Using the martingale condition (16), the process 𝑆_𝑡 is a local martingale under 𝑄 and 𝑑𝑆_𝑡=𝑆_𝑡−𝜎𝑣_𝑡𝑑𝑊^𝑄_1,𝑡+_ℜ𝑧𝑁_𝑄(𝑑𝑠,𝑑𝑧), (28) or equivalently, 𝑑𝑆_𝑡=𝑆_𝑡−𝑟_𝑡𝑑𝑡+𝜎𝑣_𝑡𝑑𝑊^𝑄_1,𝑡+_ℜ𝑧𝑁_𝑄(𝑑𝑠,𝑑𝑧). (29) Therefore the dynamic of the (𝑆,𝑣) under the risk-neutral martingale measure 𝑄 is 𝑑𝑆_𝑡=𝑆_𝑡−𝑟_𝑡𝑑𝑡+𝜎𝑣_𝑡𝑑𝑊^𝑄_1,𝑡+_ℜ𝑁_𝑄(𝑑𝑠,𝑑𝑧)𝑑𝑣_𝑡=𝑎𝑡,𝑆_𝑡,𝑣_𝑡+𝜃_1,𝑡𝑑𝑡+𝑏𝑡,𝑆_𝑡,𝑣_𝑡𝜌_𝑡𝑑𝑊^𝑄_1,𝑡+1−𝜌²_𝑡𝑑𝑊^𝑄_2,𝑡. (30)(31)

Proposition 2. Suppose that the asset return process 𝑆_𝑡 is governed by SED (30). Define 𝑋_𝑡=ln𝑆_𝑡; then under the risk-neutral measure 𝑄, 𝑑𝑋_𝑡=^𝑡₀𝑟_𝑠−12𝜎²𝑣_𝑠+𝑑𝑡_|𝑧|<1{ln(1+𝑧)−𝑧}𝜈_𝑄(𝑑𝑡,𝑑𝑧)+^𝑡₀𝜎𝑣_𝑠𝑑𝑊^𝑄_1,𝑠+_ℜln(1+𝑧)𝑁_𝑄(𝑑𝑡,𝑑𝑧). (32)

Proof. In a similar way to that of Proposition 1, the proof follows from the Itô formula applied to 𝑋_𝑡=ln(𝑆_𝑡). 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 𝑑𝑆_𝑡=𝑆_𝑡−𝜇−𝑞𝑑𝑡+√𝑣_𝑡𝑑𝑊_1,𝑡+_ℜ𝑧𝑁(𝑑𝑠,𝑑𝑧),𝑑𝑣_𝑡=𝛽𝑚−𝑣_𝑡𝑑𝑡+𝜎√𝑣_𝑡𝑑𝑊_2,𝑡, (33)(34) where 𝑚 is the rate of reversion computed by the long run mean of the process 𝑣∶=(𝑣_𝑡)_𝑡≥0 and 𝛽 is the log rate at which 𝑣 tends to 𝑚. The risk-neutral counterparts to these equations are, respectively, 𝑑𝑆_𝑡=𝑆_𝑡−𝑟𝑑𝑡+√𝑣_𝑡𝑑𝑊^𝑄_1,𝑡+_ℜ𝑧𝑁_𝑄(𝑑𝑠,𝑑𝑧),𝑑𝑣_𝑡=𝜇−𝑞−𝑟+𝛽𝑚−𝑣_𝑡𝑑𝑡+𝜎√𝑣_𝑡𝑑𝑊^𝑄_2,𝑡, (35)(36) where 𝑑𝑊^𝑄_2,𝑡 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: 𝑆_𝑡=𝑆(0)exp^𝑡₀1𝜇−𝑞−2𝑣_𝑠𝑑𝑠+^𝑡₀√𝑣_𝑠𝑑𝑊_1,𝑠+^𝑡_0|𝑧|<1{()}𝜈()+ln1+𝑧−𝑧𝑑𝑧𝑑𝑠_{𝑡0|𝑧|<1}()𝑁()+ln1+𝑧𝑑𝑠,𝑑𝑧_{𝑡0|𝑧|≥1}ln(1+𝑧)𝑁(𝑑𝑠,𝑑𝑧). (37) 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), ln(^𝑡_0|𝑧|≥11+𝑧)𝑁(𝑑𝑠,𝑑𝑧), (38) 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, ln(^𝑡_0|𝑧|<11+𝑧)𝑁(𝑑𝑠,𝑑𝑧)=lim_𝜀↓0^𝑡_{0𝜀<|𝑧|<1}ln(1+𝑧)𝑁(𝑑𝑠,𝑑𝑧) (39) 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: 𝑆^‡_𝑡,𝜀=𝑆(0)exp^𝑡₀𝑏_𝑠,𝜀𝑑𝑠+^𝑡₀√𝑣_𝑠𝑑𝑊_1,𝑠+^𝑡_{0|𝑧|≥𝜀}ln(1+𝑧)𝑁(𝑑𝑠,𝑑𝑧), (40) where 𝑏_𝑠,𝜀∶=𝜇−𝑞−(1/2)(𝑣_𝑠)−∫_{𝜀<|𝑧|≤1}{ln(1+𝑧)−𝑧}𝜈(𝑑𝑧), and 𝑆_𝑡,𝜀∶=𝑆_𝑡−𝑆^‡_𝑡,𝜀=^𝑡_0|𝑧|<1{()}𝜈()+ln1+𝑧−𝑧𝑑𝑧𝑑𝑠_{𝑡0|𝑧|<𝜀}ln(1+𝑧)𝑁(𝑑𝑠,𝑑𝑧). (41) Define 𝜎²_𝜀∶=∫_{|𝑧|≤𝜀}𝑧²𝜈(𝑑𝑧) and suppose the 𝜈 has no atoms in a neighborhood of the origin. Introduce a candidate approximation of SDE (37) in distribution sense: 𝑆^†_𝑡,𝜀=𝑆(0)exp^𝑡₀𝑏_𝑠,𝜀𝑑𝑠+^𝑡₀𝑣²_𝑠+𝜎²_𝜀𝑑𝑊_1,𝑠+^𝑡_{0|𝑧|≥𝜀}ln(1+𝑧)𝑁(𝑑𝑠,𝑑𝑧). (42) If lim_𝜀→01𝜀𝜎_𝜀=∞, (43) then the process (𝑣_𝜀⁻¹𝑆_𝑡,𝜀)_𝑡≥0 converges in distribution to a standard Brownian motion (𝐵_𝑡)_𝑡≥0, and by (43), it holds that sup_𝑥∈ℜ𝑃𝑆_𝑡≤𝑥−𝑃𝑆^†_𝑡,𝜀𝜀≤𝑥≤𝑐𝜎_𝜀. (44) Therefore, the distribution of 𝑆_𝑡 (37) (with infinite jump activity) can be approximated closely to 𝑆^†_𝑡,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 process 𝑣=(𝑣_𝑡)_{0≤𝑡≤𝑇}, where 𝑣_𝑡 satisfies the SED (36) for simulation purposes. The process 𝑣_𝑡 is obtained using stochastic calculus transformation techniques [10, Theorem  1]. Let 𝑉_𝑡=𝐹(𝑡,𝑣_𝑡)∈𝐶^1,2(ℜ). Applying Itô’s lemma to the process 𝑉, we obtain the following equation: 𝑑𝑉_𝑡=𝐹_𝑡𝑡,𝑣_𝑡𝑑𝑡+𝐹_𝑣𝑡,𝑣_𝑡𝑑𝑣_𝑡+12𝐹_𝑣𝑣𝑡,𝑣_𝑡𝜎²_𝑣𝑣_𝑡𝑑𝑡. (45) Then the solution of the stochastic process 𝑣 is 𝑣_𝑡=𝑒^−𝛽𝑉_𝑡2², (46) where 𝑉_𝑡=2√𝑣₀+2𝐼_𝑔(𝑡),𝐼_𝑔(𝑡)=12^𝑡₀𝑒^𝛽/2𝛽𝑚−𝜎²/4√𝑣_𝑠𝑑𝑠+𝜎𝑑𝑊^𝑄_2,𝑠. (47) 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 [𝑡_𝑛,𝑡_𝑛−1] where 𝑡_𝑛=𝑛Δ. Let 𝑣_𝑛 denote 𝑣_𝑡𝑛. 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 solution 𝑣 at time 𝑡_𝑛+1 is generated by the following reclusive formula: 𝑣_𝑛+1=𝑣₀+2√𝑣_𝑛𝐼_𝑔̃𝑡_𝑛+𝐼²_𝑔̃𝑡_𝑛, (48) where 𝐼_𝑔̃𝑡_𝑛=12𝑒^𝛽/2𝛽𝑚−𝜎²/4Δ_𝑛√𝑣_𝑛+Δ𝑊^𝑄_2,𝑛,Δ𝑊^𝑄_2,𝑛=^𝑡𝑛+1_𝑡𝑛𝑑𝑊^𝑄_2,𝑡=𝑊^𝑄_2,𝑡𝑛+1−𝑑𝑊^𝑄_2,𝑡𝑛 (49) is the 𝒩(0,Δ_𝑛) distributed increment of the Brownian motion 𝑊 on [𝑡_𝑛,𝑡_𝑛+1] and Δ_𝑛=^𝑡𝑛+1_𝑡𝑛𝑑𝑡=𝑡_𝑛+1−𝑡_𝑛 (50) is the length of the time discretization subinterval [𝑡_𝑛,𝑡_𝑛+1].

For numerical experiments, we simplified implementation to generate processes 𝑣 that only apply explicit schemes to (48).

The discretized version of the risk-neutral log return process 𝑆^†_𝑡,𝜀 is deduced from the approximation of each component as follows: let Δ>0 be given and let 𝑋^†_𝑡∶=log𝑆^†_𝑡,𝜀. Rewrite the 𝑆^†_𝑡,𝜀 for log returns: 𝑋^†_𝑡+Δ=𝑋^†_𝑡+_𝑡^𝑡+Δ𝑏_𝑠,𝜀𝑑𝑠+_𝑡^𝑡+Δ𝑣²_𝑠+𝑣²_𝜀𝑑𝑊_1,𝑠+_𝑡^𝑡+Δ_{|𝑧|≥𝜀}ln(1+𝑧)𝑁(𝑑𝑠,𝑑𝑧). (51) 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 _𝑡^𝑡+Δ𝑏_𝑠,𝜀𝑑𝑠≈𝑏_𝑡,𝜀Δ. (52) Following the first integral term, we get _𝑡^𝑡+Δ𝑣²_𝑠+𝑣²_𝜀𝑑𝑊_1,𝑠≈𝑣²_𝑡+𝑣²_𝜀Δ𝑊_1,𝑡, (53) with Δ𝑊_1,𝑡∶=𝑊_1,𝑡+Δ−𝑊_1,𝑡. From the remaining integral in (51), we get ln(_𝑡^𝑡+Δ_{|𝑧|≥𝜀}1+𝑧)Δ𝑁(𝑑𝑠,𝑑𝑧)≈Δ_𝑁𝑆^†_𝑡,𝜀Δ, (54) where Δ_𝑁𝑆^†_𝑡,𝜀 is the jump process of 𝑆^†_𝑡,𝜀 caused by the jump of 𝑁(𝑡,𝑧), as denoted by Δ_𝑁𝑆^†_𝑡,𝜀=_{|𝑧|≥𝜀}ln(1+𝑧)𝑁(𝑡,𝑑𝑧). (55) 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 process 𝑋_𝑡 is displayed in the following proposition.

Proposition 4. Based on Euler approximation, the discretization scheme for 𝑆^†_𝑡,𝜀 is as follows: 𝑆^†_𝑛+1,𝜀=𝑆^†_𝑛,𝜀exp𝑏_𝑛,𝜀Δ_𝑛+𝑣²_𝑛+𝑣²_𝜀Δ_𝑛𝑍_𝑛+1+Δ_𝑁𝑆^†_𝑛,𝜀Δ_𝑛 (56) with 𝑣_𝑛 following from the scheme (48).

Proof. We use the approximation of (52), (53), and (54) to approximate 𝑆^†_𝑡+Δ,𝜀. With the transformation 𝑆^†=exp(𝑋^†), the discretized version of the risk-neutral 𝑆^†_𝑡+Δ,𝜀 is 𝑆^†_𝑡+Δ,𝜀≈𝑆^†_𝑡,𝜀exp𝑏_𝑡,𝜀Δ+𝑣²_𝑡+𝑣²_𝜀Δ𝑊_1,𝑡+Δ_𝑁𝑆^†_𝑡,𝜀Δ. (57) We apply this recursively at any Δ=Δ_𝑛∶=𝑡_𝑛+1−𝑡_𝑛>0 and replace the increment of Brownian motion 𝑊 with √Δ𝑍,  𝑍∼𝒩(0,1). More explicitly, we have (56).

From (56) and setting 𝑋^†_𝑡,𝜀∶=log𝑆^†_𝑡+Δ,𝜀, we have 𝑋^†_𝑡+Δ,𝜀≈𝑋^†_𝑡+𝑏_𝑡,𝜀Δ+𝑣²_𝑡+𝑣²_𝜀Δ𝑊_1,𝑡+Δ_𝑁𝑆^†_𝑡,𝜀Δ. (58) 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,𝜎_𝑌). The remaining parameters 𝑞 are fixed that calibrate the parameter based on historical data.

The conditionals of the process 𝑋^† are normal and independent, due to the locally constancy of the SV 𝑣 on small Δ𝑡. The log-likelihood function for the asset return 𝑋^†_𝑛 and 𝑣_𝑛 is then given by ℒ(Θ)=^𝑁0_𝑛=1log𝑓𝑋^†_𝑛,𝑣_𝑛;Θ. (59) The probability density function (pdf) of process 𝑋_𝑛 given 𝑋_𝑛−1 together with the volatility 𝑣_𝑛 given 𝑣_𝑛−1 has a conditional Gaussian distribution 𝑓𝑥^†_𝑛+1,𝑣_𝑛+1∣Θ=^+∞_𝑗=0ℚ𝑛_𝑡=𝑗×𝑓_𝒩𝑥^†_𝑛+1,𝑣_𝑛+1;𝑏_𝑛,𝜀Δ_𝑛+𝑥^†_𝑛+𝑗𝜇_𝑌,𝑣²_𝑛Δ_𝑛+𝑗𝜎²_𝑌, (60) where 𝑣_𝑛∶=𝑣²_𝑛+𝑣²_𝜀 and 𝑓_𝒩𝑥^†_𝑛+1,𝑣_𝑛+1=1;Θ2𝜋𝑣²_𝑛+𝑗𝜎²_𝑌−𝑥exp^†𝑛+1−𝑏_𝑛,𝜀−𝑥^†_𝑛−𝑗𝜇_𝑌²2𝑣²_𝑛+𝑗𝜎²_𝑌. (61) This pdf is the sum of the conditional probability densities weighted by the probability of the conditioning variables of the number of jumps 𝑛_𝑡=𝑁_𝑡−𝑁_{𝑡−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 𝑓𝑥^†_𝑛+1=;Θ1−𝜆𝑡_𝑛+1𝑓_𝒩𝑥^†_𝑛+1,𝑣_𝑛+1;𝑏_𝑛,𝜀Δ_𝑛+𝑥^†,𝑣²_𝑛Δ_𝑛+𝜆𝑡_𝑛+1𝑓_𝒩𝑥^†_𝑛+1,𝑣_𝑛+1;𝑏_𝑛,𝜀Δ_𝑛+𝑥^†_𝑛+𝜇_𝑌,𝑣²_𝑛Δ_𝑛+𝜎²_𝑌. (62)

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 (𝛼_𝑡)_{𝑡∈[0,𝑇]} be a continuous-time Markov chain on finite space 𝒮∶={1,2,…,𝐾} and the transition rate matrix 𝑄=[𝑞_𝑖𝑗] where the transition rate 𝑞_𝑖𝑗 is calculated with respect to the natural filtration generated by continuous-time Markov chain 𝛼, ℱ^𝛼_𝑡∶={𝜎(𝛼_𝑠)∶0≤𝑠≤𝑡} and given by 𝑞_𝑖𝑗𝑞≥0if𝑖≠𝑗∀𝑖,𝑗∈𝒮,_𝑖𝑖=−^𝐾_{𝑗=0,𝑗≠𝑖}𝑞_𝑖𝑗otherwise. (63) By definition, the Markov chains 𝛼_𝑡 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, ℙ𝛼_𝑡+𝑠=𝑗∣𝛼_𝑠=𝑖=𝑞_𝑖𝑗=ℙ𝛼_𝑡=𝑗∣𝛼₀=𝑖 (64) for all 𝑠,𝑡≥0. The regime switching (RS) of SVL model (35)-(36) is defined by the following SDE: 𝑑𝑆_𝑡=𝑆_𝑡−𝑟𝛼_𝑡𝑑𝑡+√𝑣_𝑡𝑑𝑊^𝑄_1,𝑡+_ℜ𝑧𝑁_𝑄(𝑑𝑠,𝑑𝑧),𝑑𝑣_𝑡=𝜇𝛼_𝑡−𝑞𝛼_𝑡−𝑟𝛼_𝑡+𝛽𝛼_𝑡𝑚𝛼_𝑡−𝑣_𝑡𝛼𝑑𝑡+𝜎_𝑡√𝑣_𝑡𝑑𝑊^𝑄_2,𝑡. (65)(66) The processes 𝑟(𝛼_𝑡), 𝜇(𝛼_𝑡), 𝑞(𝛼_𝑡), 𝛽(𝛼_𝑡), 𝑚(𝛼_𝑡), and 𝜎(𝛼_𝑡) are constants taking values, respectively, in 𝑟(𝒮), 𝜇(𝒮), 𝑞(𝒮), 𝛽(𝒮), 𝑚(𝒮), and 𝜎(𝒮) such that 𝑟(𝒮)∶=𝑟₁,…,𝑟_𝐾,𝜇(𝒮)∶=𝜇₁,…,𝜇_𝐾,𝑞(𝒮)∶=𝑞₁,…,𝑞_𝐾,𝛽(𝒮)∶=𝛽₁,…,𝛽_𝐾,𝑚(𝒮)∶=𝑚₁,…,𝑚_𝐾,𝜎(𝒮)∶=𝜎₁,…,𝜎_𝐾. (67) The solution 𝑆_𝑡(𝛼_𝑡) of (65) with SV (66) is called an RS of SVGL model. Let ℱ^𝑆_𝑛 be the filtration of historical value of the process 𝑆_𝑛 untial time 𝑡_𝑛. By Proposition 4, its approximation of the solution 𝑆_𝑡(𝛼_𝑡) in discrete version is given by 𝑆^†_𝑛+1,𝜀=𝑆^†_𝑛,𝜀exp𝑏_𝑛,𝜀𝛼_𝑛Δ_𝑛+𝑣²_𝑛𝛼_𝑛+𝑣²_𝜀Δ_𝑛𝑍_𝑛+1+Δ_𝑁𝑆^†_𝑛,𝜀Δ_𝑛. (68)

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 states 𝛼_𝑘=𝑘,  𝑘=1,2, the conditional distribution density of 𝑋^†_𝑛+1,𝜀, the log price of 𝑆^†_𝑛+1,𝜀, has associated parameters across regimes given by 𝑓𝑋^†_𝑛+1,𝜀∣ℱ^𝛼_𝑛;Θ=𝑓𝑋^†_𝑛+1,𝜀∣ℱ^𝛼_𝑛,𝛼_𝑛=1;Θ₁𝑞_1,𝑛+𝑓𝑋^†_𝑛+1,𝜀∣ℱ^𝛼_𝑛−1,𝛼_𝑛=2;Θ₂1−𝑞_1,𝑛 (69) up to the conditional probability 𝑞_1,𝑛=𝑄{𝛼_𝑛=1∣ℱ^𝛼_𝑛}.

Assume that a Markov switching or jump process is independent of Brownian motion 𝑊^𝑄₁,𝑊^𝑄₂. Combining both regime switching and jumps with the locally constant of the SV process 𝑣, with conditional density of 𝑋^†_𝑛,𝜀, from (62), we get 𝑓𝑋^†_𝑛+1,𝜀∣ℱ^𝑋†_𝑛+1,𝛼_𝑛=𝑘;Θ_𝑘=1−𝜆𝑡_𝑛+1𝑓𝑆^†_𝑛,𝜀∣ℱ^𝑋†_𝑛,𝛼_𝑛=𝑗,nojump;Θ_𝑘+𝜆𝑡_𝑛+1𝑓𝑋^†_𝑛,𝜀∣ℱ^𝑋†_𝑛−1,𝛼_𝑛=𝑖,jump;Θ_𝑘=1−𝜆𝑡_𝑛+1𝑓_𝒩𝑥^†_𝑛,𝑣_𝑛;𝑏_𝑛,𝜀Δ_𝑛+𝑥_𝑛,𝑣²_𝑛Δ_𝑛+𝜆𝑡_𝑛+1𝑓_𝒩𝑥^†_𝑛,𝑣_𝑛;𝑏_𝑛,𝜀Δ_𝑛+𝑥_𝑛+𝑗𝜇_𝑌,𝑣²_𝑛Δ_𝑛+𝜎²_𝑌. (70) Substituting (70) into (69) gives the conditional density of 𝑋^†_𝑛,𝜀 used in constructing the likelihood function for the data.

The set of the parameters Θ 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⁾∶=(𝑟⁰_𝑘,𝜎⁰_𝑘,𝑚⁰_𝑘,𝛽⁰_𝑘,𝜇⁰_𝑌,𝑘,𝜎⁰_𝑌,𝑘) 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 𝑣_𝑛 of (48) as a variance process of discretization for approximation process 𝑋_𝑛,𝜀.(3)Generate the process 𝑋_𝑛,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 fix 𝑟 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 𝜇_𝑌=0.2250 and 𝜎_𝑌=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 Algorithm 5. The log prices 𝑋^†_𝑡,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(𝑋_𝑡,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.

(a)

(b)

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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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.