Discrete Dynamics in Nature and Society

Volume 2016, Article ID 1580941, 15 pages

http://dx.doi.org/10.1155/2016/1580941

## Modeling Financial Time Series Based on a Market Microstructure Model with Leverage Effect

^{1}Hunan Province Higher Education Key Laboratory of Power System Safety Operation and Control, Changsha University of Science and Technology, Changsha, Hunan 410004, China^{2}School of Information Science & Engineering, Central South University, Changsha, Hunan 410083, China^{3}Hunan Engineering Laboratory for Advanced Control and Intelligent Automation, Changsha, Hunan 410083, China

Received 31 August 2015; Accepted 21 December 2015

Academic Editor: Filippo Cacace

Copyright © 2016 Yanhui Xi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The basic market microstructure model specifies that the price/return innovation and the volatility innovation are independent Gaussian white noise processes. However, the financial leverage effect has been found to be statistically significant in many financial time series. In this paper, a novel market microstructure model with leverage effects is proposed. The model specification assumed a negative correlation in the errors between the price/return innovation and the volatility innovation. With the new representations, a theoretical explanation of leverage effect is provided. Simulated data and daily stock market indices (Shanghai composite index, Shenzhen component index, and Standard and Poor’s 500 Composite index) via Bayesian Markov Chain Monte Carlo (MCMC) method are used to estimate the leverage market microstructure model. The results verify the effectiveness of the model and its estimation approach proposed in the paper and also indicate that the stock markets have strong leverage effects. Compared with the classical leverage stochastic volatility (SV) model in terms of DIC (Deviance Information Criterion), the leverage market microstructure model fits the data better.

#### 1. Introduction

Financial system is a complex dynamic system. Many mathematical models have already been proposed to describe the dynamics of financial markets. With the fast development of mathematical finance in recent years, stochastic differential equations have been widely used to describe the dynamics of a wide variety of random in finance. The current most acceptable theorem of price movements in financial markets is that they are random walks with predictive errors close to white noise and this is due to the Markov properties of financial time series. Hence, much of financial theory is based on this assumption.

It has been long recognized that the returns of financial assets are negatively correlated with changes in the volatilities of returns; that is, there is leverage effect between volatility and price/return. Black [1] and Christie [2] have found empirical evidence of the leverage effect; that is, volatility tends to rise in response to bad news but fall in response to good news. Christie [2] provides a theoretical explanation of leverage effect under a Modigliani/Miller economy.

Modeling of financial market volatility has been one of the most active areas of research in empirical finance and time series econometrics over the past two decades. Two types of volatility models, the ARCH (autoregressive conditional heteroskedasticity) model and the SV (stochastic volatility) model, are well estimated in financial econometrics. Numerous researchers develop their extensions and found overwhelming evidence of leverage effect. For instance, Bollerslev [3] proposed the generalized ARCH (GARCH) model, which formulates the serial dependence of volatility and incorporates the past observations into the future volatility. Nelson [4] proposed the EGARCH (exponential GARCH) specification, modeling the leverage effect, which refers to the increase in volatility following a previous drop in stock returns. while Glosten et al. [5] developed a threshold indicator function GARCH model with leverage effect, which is commonly called the GJR model. A common idea used in the above models is the leverage effect, in which negative shocks to price/return increase the predictable volatility to a greater extent than do positive shocks.

On the other hand, regarding volatility clustering, the stochastic volatility (SV) model has been widely used to model the time-varying variance of time series in financial econometrics. In the SV literature, the asymmetric (leverage) volatility response is often studied by specifying a negative correlation between the return innovation and the volatility innovation. This classical leverage SV model, proposed by Harvey and Shephard [6], requires correlation between the return innovation and the volatility innovation . Instead of the dependence, that is, , Jacquier et al. [7] generalized the basic SV model with leverage effect by allowing for an dependence between and ; that is, . Yu [8] gives comments on the JPR specification [7] and shows that it does not necessarily lead to a leverage effect and hence is not theoretically justified. Following Harvey and Shephard [6], Jacquier et al. [7], Yu [8, 9], Omori et al. [10], Wang et al. [11], Asai and McAleer [12], and Tsiotas [13] extend the SV specification with leverage.

Differing from the ARCH model and the SV model, a phenomenological model based on identifying the processes influencing the demand and supply of a market was proposed, which was called the market microstructure model [14]. O’Hara [15] defines market microstructure as “the study of the process and outcomes of exchanging assets under a specific set of rules. While much of economics abstracts from the mechanics of trading, microstructure theory focuses on how specific trading mechanisms affect the price formation process.” In microeconomics, supply and demand is an economic model of price determination in a market; see, for example, Marshall [16], Zhang [17]. On the other hand, the market liquidity affects the asset prices and expected returns. Theory and empirical evidence suggests that, for an asset with given cash flow, the higher its market liquidity, the lower its expected return (e.g., [18, 19]). In this paper, the market microstructure model assumes that asset price is driven by the excess demand, and the amplitude of price changes is dependent on the liquidity of the market. Some improvements of the market microstructure model and their estimation approaches as well as applications [20, 21] were also presented. Although Peng et al. [22, 23] and Xi et al. [24, 25] proposed the generalized market microstructure (GMMS) models, which included jump component for capturing the low-frequency and large-amplitude abnormal vibrations of price, they did not consider the important property, namely, leverage effect. Recently, to explain essential characteristics of skewness and heavy tails, Xi et al. [25] proposed the heavy-tailed market microstructure model based on Student- distribution (MM-). However, all of the above market microstructure models are difficult to explain the asymmetry in the relation between volatility and price/return. Motivated by the empirical evidence, this paper is concerned with the specification for modeling financial leverage effect in the market microstructure model, which allows for a dependence between the price innovation and the volatility innovation to pick up the kind of leverage behavior.

The remainder of the study is organized as follows. Section 2 introduces the market microstructure model with leverage and gives a theoretical explanation of leverage effect. In Section 3, we discuss the MCMC estimation for our market microstructure model with leverage. Section 4 describes the simulated data results by MCMC method. Section 5 displays and discusses the empirical findings using the leverage market microstructure model. Section 6 provides the model comparison between the leverage SV model and the leverage market microstructure model. Section 7 gives some concluding remarks.

#### 2. Leverage in the Market Microstructure Model

To deal with the dynamics of a financial market from different points of view, one phenomenological microstructure model based on identifying different processes influencing the demand and supply of the market is defined by [26]where is the asset price, is the (inverse of) market liquidity, and is the excess demand which is defined by , where is the instantaneous demand and is the instantaneous supply at any given instant of time for the asset. characterizes whether the market is overvalued (, which tends to push the price up) or undervalued (, which tends to push the price down). Model (1) assumes that price is driven by the excess demand , and the amplitude of price changes is determined by the market liquidity, that is, .

However, model (1) only provides an abstract description for the dynamics of market. To estimate the two hidden market state variables ( and ) from a time series of price, model (1) can then be written as [14, 20] where , , and are independent Brownian motion processes and , , , , and and are constant parameters.

The first equation in model (2) describes the financial asset price process. It is notable that the conditional expected value and the conditional variance of the financial asset price process are given by

The second equation in model (2) models the process of the immeasurable hidden excess demand variable . The third equation in model (2) models the dynamics of liquidity, which together with the first equation resembles a stochastic volatility (SV) model in which the variance is specified to follow some latent stochastic process. Consequently, the third equation reveals characterization of volatility dynamics. In contrast with the SV model and the ARCH model, the market microstructure model (model (2)) offers a better representation of the internal characteristics of a financial price varying process and information which possess better stability than the market trend information obtained for the mere prediction of a price process.

To make the above market microstructure model characterize the leverage effect for financial variables, here we assume that , are two dependence Brownian motions with correlation measured by ; that is, . Here, the correlation coefficient implies model (2) with leverage effect, while implies model (2) without leverage effect.

In the empirical literature, the continuous model is often discretized to facilitate estimation. In mathematics, the Euler-Maruyama method [27] is a simple and generalized method for the approximate numerical solution of a stochastic differential equation.

A general form for a stochastic differential equation iswhere stands for the Wiener process that depends continuously on .

Stochastic differential equation (4) can be written in integral form asTo construct a numerical method from the integral form (5), we begin by setting for some positive integer and and also define the numerical approximation to as . Setting successively and in integral form (5), we can obtainIf we subtract (7) from (6), then we obtainWe can now consider approximating each of the integral terms. For the first integral in (8), we can use the conventional deterministic quadrature:And, for the second integral, we use the Ito formula:By combining these together, the Euler-Maruyama method takes the formwhere ().

Using the above Euler-Maruyama discrete time approximation, model (2) can be then derived as follows:where , , and are iid (the abbreviation iid refers to independent and identically distributed) and , are dependent errors with correlation measured by ; that is, . is a new constant parameter, which describes the relation between , and .

To build the observation equation, is chosen as observation variable. So, the state space representation of model (12) is given bywhere is iid .

To fully understand the leverage effect, it is convenient to adopt the Gaussian nonlinear state space form with uncorrelated error terms. To do so, let us denote and rewrite (12) aswhere is iid and , , and are independent Gaussian white noise processes.

Also, the system of (14) can be alternatively specified byFor simplicity, we setIn finance, the volatility of asset price corresponds to standard deviation. Consequently, in (16) represents the volatility of asset price, which is explained by the liquidity process.

To compute the partial derivative , we first definewhere is a compounded return.

According to the derivation rule of the implicit function, we have

Obviously, if and holding everything else constant, a fall in the stock return () leads to an increase of future expected volatility . Instead, a rise in the stock return () leads to an decrease of future expected volatility . This is a so-called leverage effect, a negative shock to returns that increases the predictable volatility to a greater extent than does a positive shock.

#### 3. Estimation of the Leverage Market Microstructure Model

For estimation of the basic market microstructure model, Peng et al. [20] proposed the extended Kalman filter (EKF) and the maximum likelihood method. However, its state equations are highly nonlinear, the EKF may give a high estimation error, and it may lead to inaccurate maximum likelihood estimates. Recently, Markov Chain Monte Carlo (MCMC) methods have become one of the most important tools for estimating stochastic volatility models since it was introduced in Jacquier et al. [28] to analyze the basic SV model. A fast and reliable MCMC algorithm, proposed by Kim et al. [29], was used in the SV model (see e.g., [10, 30, 31]). On the other hand, Bauwens and Lubrano [32], Vrontos et al. [33], and Nakatsuma [34] develop the MCMC estimation method for the models in the GARCH class. Their estimation results are very good since MCMC provides a fully likelihood-based inference.

In statistics, Markov Chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the desired distribution. To facilitate an efficient posterior inference using WinBUGS (Windows version of Bayesian Analysis Using Gibbs Sampler, a statistical software for Bayesian analysis using MCMC methods. WinBUGS Homepage: http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/contents.shtml), the following state and observation equations for the model (13) were obtained as

In this context, we face the problem of simulating a multivariate density consisting of the parameter and the unobserved states , , and . Let be the observed asset and the vectors are unobserved states, where , , and . By Bayes’ theorem, the joint posterior distribution of the unobservable variables given the data is obtained aswhere the joint prior density isand the likelihood function is

A Markov chain is generated by Gibbs sampling and its stationary distribution is the joint posterior distribution (21). Under a given prior probability density for , it is now possible to efficiently sample the posterior density by the MCMC technique. We develop the following sampling algorithm for the model (13).

*Step 1. *Initialize

*Step 2. *Sample from

*Step 3. *Sample from

*Step 4. *Sample from

*Step 5. *Sample from

*Step 6. *Sample from

*Step 7. *Sample from

*Step 8. *Sample from

*Step 9. *Sample from

*Step 10. *Sample from

*Step 11. *For , sample from where .

*Step 12. *For , sample from where .

*Step 13. *For , sample from where .

*Step 14. *Repeat the above procedure and stop until the number of iterations is , which must be large enough to guarantee that the Markov chain reaches stationary distribution.

#### 4. Simulation Study

For simulation study of the proposed leverage market microstructure model by the MCMC algorithm, 1000 observations from model (13) are generated with the unknown parameters , , , , , , and and the known parameters , . The initial value , , and are taken as 199.3, , and 0.1024, respectively. Figures 1 and 2 show the simulation data made from model (13).