Markov chain Monte Carlo methods for stochastic volatility models

Abstract

This paper is concerned with simulation-based inference in generalized models of stochastic volatility defined by heavy-tailed Student-t distributions (with unknown degrees of freedom) and exogenous variables in the observation and volatility equations and a jump component in the observation equation. By building on the work of Kim, Shephard and Chib (Rev. Econom. Stud. 65 (1998) 361), we develop efficient Markov chain Monte Carlo algorithms for estimating these models. The paper also discusses how the likelihood function of these models can be computed by appropriate particle filter methods. Computation of the marginal likelihood by the method of Chib (J. Amer. Statist. Assoc. 90 (1995) 1313) is also considered. The methodology is extensively tested and validated on simulated data and then applied in detail to daily returns data on the S&P 500 index where several stochastic volatility models are formally compared under different priors on the parameters.

Introduction

Stochastic volatility models have gradually emerged as a useful way of modeling time-varying volatility with significant potential for applications, especially in finance (Taylor, 1994; Shephard, 1996; Ghysels et al., 1996) for a discussion of the models and the related literature). In this paper we consider two versions of SV models and extend existing work on more restricted models to develop efficient and fast Bayesian Markov chain Monte Carlo (MCMC) estimation algorithms. We develop a straightforward procedure for computing the marginal likelihood and Bayes factors for SV models. This procedure combines a simulation-based filter for estimating the likelihood ordinate with the method of Chib (1995) for estimating the posterior density of the parameters. The marginal likelihood procedure is tested on several problems and it is shown that the approach is capable of correctly choosing between various competing models. As a by-product of our work, we provide a method for filtering the current value of the unobserved volatility using contemporaneous data. This method is a simpler alternative to the reprojection method proposed by Gallant and Tauchen (1998).

The simplest formulation of the SV model, labelled SV₀, is given by

$$\text{y}{}_{\mathrm{t}}\text{=}\text{exp}\text{(h}{}_{\mathrm{t}}\text{/2)u}{}_{\mathrm{t}}\text{,h}{}_{\mathrm{t}}\text{=μ+φ(h}{}_{\mathrm{t-1}}\text{−μ)+ση}{}_{\mathrm{t}}\text{,}\text{t⩽n,}$$

where y_t is the response variable, h_t is the unobserved log-volatility of y_t and the errors u_t and η_t are Gaussian white noise sequences. This model has been heavily analyzed in the literature. The first Bayesian analysis was provided by Jacquier et al. (1994) where the posterior distribution of the parameters was sampled by MCMC methods using the distributions h_t|y,h_(−t),μ,φ,σ(t⩽n); μ|y,h,φ,σ; φ|y,h,μ,σ and σ|y,h,μ,φ, where h=(h₁,…,h_n) and h_(−t) denotes all the elements of h excluding h_t. Although this algorithm is conceptually simple it is not particularly efficient from a simulation perspective, as is shown by Kim et al. (1998) who develop an alternative, more efficient, MCMC algorithm for the above model. The efficiency gain in the Kim, Shephard and Chib algorithm arises from the joint sampling of (φ,σ) in one block marginalized over both {h_t} and μ, followed by the sampling of {h_t} in one block conditioned on everything else in the model.

In this paper, we are concerned with two extensions of the basic SV model. The first of these models, which we label the SVt model, is defined by Student-t observation errors, level effect in the volatility and covariates in the volatility evolution. The model is given by the specification

$$\text{y}{}_{\mathrm{t}}\text{=x}{}_{\mathrm{t}}\text{′β+w}{}_{\mathrm{t}}{}^{\mathrm{\gamma }}\text{exp}\text{(h}{}_{\mathrm{t}}\text{/2)u}{}_{\mathrm{t}}\text{,h}{}_{\mathrm{t}}\text{=μ+z}{}_{\mathrm{t}}\text{′α+φ(h}{}_{\mathrm{t-1}}\text{−μ)+ση}{}_{\mathrm{t}}\text{,}\text{t⩽n,}$$

where x_t, w_t and z_t are covariates, γ denotes the level effect and u_t is distributed as a Student-t random variable with mean zero, variance ν/(ν−2) and ν>2 degrees of freedom. By exploiting the well-known fact that the Student-t distribution can be expressed as a particular scale mixture of normals, we write u_t=λ_t^−1/2ε_t where ε_t is standard normal N(0,1) and λ_t is i.i.d. Gamma(ν/2,ν/2). In the SV context, Student-t error-based models were used by Harvey et al. (1994), while Mahieu and Schotman (1998) discuss the use of a mixture distribution. Recently, Jacquier et al. (1999) have computed the posterior density of the parameters of a Student-t-based SV model. We assume that the degrees of parameter ν of the t-distribution is unknown and is estimated from the data. Finally, {w_t} is a non-negative process, such as the lag of the interest rate (see, for example, Andersen and Lund (1997) and the references contained within).

As motivation we should mention that the above model can be thought of as an Euler discretization of a Student-t-based Lévy process with additional stochastic volatility effects. The latter models are being actively studied in the continuous-time mathematical options and risk assessment literature. Leading references include Eberlein (2001), Prause (1999) and Eberlein and Prause (2002), while an introductory exposition is given in Barndorff-Nielsen and Shephard (2002, Chapter 2). The extension to allow for stochastic volatility effects is discussed in Eberlein and Prause (2002) and Eberlein et al. (2001).

The second model we discuss is similar to the SVt model except that it contains a jump component in the observation equation to allow for large, transient movements. This model, which we call the SVt plus jumps model (SVJt), is defined as

$$\text{y}{}_{\mathrm{t}}\text{=x}{}_{\mathrm{t}}\text{′β+k}{}_{\mathrm{t}}\text{q}{}_{\mathrm{t}}\text{+w}{}_{\mathrm{t}}{}^{\mathrm{\gamma }}\text{exp}\text{(h}{}_{\mathrm{t}}\text{/2)u}{}_{\mathrm{t}}\text{,h}{}_{\mathrm{t}}\text{=μ+z}{}_{\mathrm{t}}\text{′α+φ(h}{}_{\mathrm{t-1}}\text{−μ)+ση}{}_{\mathrm{t}}\text{,}\text{t⩽n,}$$

where q_t is a Bernoulli random variable that takes the value one with unknown probability κ and the value zero with probability 1−κ. The time-varying variable k_t represents the size of the jump when a jump occurs and is assumed to a priori follow the distribution

$$\text{log}\text{(1+k}{}_{\mathrm{t}}\text{)∼}\text{N}\text{(−0.5δ}{}^2\text{,δ}{}^2\text{),}$$

following Andersen et al. (2002). Taken together k_tq_t can be viewed as a discretization of a finite activity Lévy process. Jump models are quite popular in continuous time models of financial asset pricing (for example, Merton, 1976; Ball and Torous, 1985; Bates, 1996; Duffie et al., 2000; Barndorff-Nielsen and Shephard, 2001a). Recent econometric work on jump-type SV models includes Barndorff-Nielsen and Shephard (2001b), Chernov et al. (2000) and Eraker et al. (2002). Our innovation is to provide an efficient and complete Bayesian tool-kit for parameter estimation, model checking, volatility estimation via filtering and model comparison.

The rest of the paper is organized as follows. In Section 2 we suggest an MCMC approach for fitting the SVt model and discuss methods for doing filtering and model choice. Section 3 presents a similar analysis for the SVJt model. Section 4 reports on an extensive Monte Carlo experiment in which the proposed estimation methods and model choice criterion are tested and validated. Then, the methodology is applied in detail to daily returns data on the S&P 500 index where several stochastic volatility models are formally compared under different priors on the parameters. Concluding remarks are presented in Section 5.