Markov chain Monte Carlo methods for stochastic volatility models
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 bywhere yt is the response variable, ht is the unobserved log-volatility of yt and the errors ut and ηt 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 ht|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,…,hn) and h(−t) denotes all the elements of h excluding ht. 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 {ht} and μ, followed by the sampling of {ht} 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 specificationwhere xt, wt and zt are covariates, γ denotes the level effect and ut 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 ut=λt−1/2εt where εt is standard normal N(0,1) and λt 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, {wt} 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 aswhere qt 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 kt represents the size of the jump when a jump occurs and is assumed to a priori follow the distributionfollowing Andersen et al. (2002). Taken together ktqt 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.
Section snippets
SVt model and Bayesian inference
A key feature of the SVt model (2) is that its likelihood function is not available easily. To see this difficulty, let ψ=(μ,φ,σ,α,γ,β,ν) denote the parameters of the model. Then, by the law of total probability, it follows that the density of the data y=(y1,…,yn) given ψ can be expressed aswhere denotes the history of the observation sequence up to time (t−1). For model (2)is the Student-t density
SVt plus jumps model
We now turn to an analysis of the SVJt model given in , . We show how the methods described in the previous section can be adapted for this setting. Recall that the model is specified aswhere qt is a Bernoulli random variable that takes the value 1 with probability κ and the value 0 with probability 1−κ and the jump size kt is distributed as log(1+kt)∼N(−0.5δ2,δ2).
To deal with this model, it is necessary to think about suitable prior
Examples
In this section, we present empirical results based on both simulated and real datasets. The purpose of these examples is to illustrate the efficacy of the algorithm along several dimensions. First, we measure the observed serial correlation in the sampled output. This is summarized by the inefficiency factors of the estimation of the posterior mean. Recall the inefficiency factor is defined aswhere ρ(k) is the autocorrelation at lag k for the parameter of interest. In Geweke (1992)
Concluding remarks
This paper has considered a class of generalized stochastic volatility models defined by heavy tails, a level effect on the volatility and covariate effects in the observation and evolution equations and a jump component in the observation equation. Two fast and efficient MCMC fitting algorithms have been developed for such models. The discussion has also considered the estimation of the volatility process and the comparison of alternative models via the marginal likelihood/Bayes factor
Acknowledgements
We thank the journal's three reviewers for their comments on previous drafts. Neil Shephard's research is supported by the UK's ESRC through the grant “Econometrics of trade-by-trade price dynamics”, which is coded R00023839.
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