Chapter on Bayesian Inference for Stochastic Volatility Modeling

This chapter reviews the major contributions over the last two decades to the literature on the Bayesian analysis of stochastic volatility (SV) models (univariate and multivariate). Bayesian inference is performed by tailoring Markov chain Monte Carlo (MCMC) or sequential Monte Carlo (SMC) schemes that take into account the specific modeling characteristics. The popular univariate stochastic volatility model with first order autoregressive dynamics (SV) is introduced in Section 1, which provides a detailed explanation of efficient MCMC and SMC algorithms. We briefly describe several extensions to the basic SV model that allows for fat-tailed, skewed, correlated errors as well as jumps (Markovian or not, smooth or not) in both observation and volatility equations, and the leverage effect via correlated errors. Multivariate SV models are presented in Section 2 with particular emphasis on Wishart random processes, cholesky stochastic volatility models and factor stochastic volatility models. Section 3 contains several illustrations of both univariate and multivariate SV models based on both MCMC and SMC algorithms. Section 4 concludes the chapter. 1 Univariate SV models Univariate stochastic volatility (SV) asset price dynamics results in the movements of an equity index St and its stochastic volatility vt via a continuous time diffusion by a Brownian motion (Rosenberg, 1972, Taylor, 1986, Hull and White, 1987, Ghysels, Harvey and Renault, 1996, Johannes and Polson, 2010): d logSt = μdt+ √ vtdB P t (1) d log vt = κ(γ − log vt)dt+ τdB t (2) where the parameters governing the volatility evolution are (μ, κ, γ, τ) and Brownian motions (B t , B V t ) possibly correlated. One extension of the above model is the stochastic volatility jump (SVJ) model that includes the possibility of jumps to asset prices. Here the equity index