Fitting and comparing stochastic volatility models through Monte Carlo simulations

Stochastic-variance models are important in describing and forecasting time-varying volatilities of financial time series. The introduction of jump components, in both the returns and the volatility process, improves the fit to the data. The goal of this paper is to examine the effectiveness of Markov Chain Monte Carlo methods in making inferences on different stochastic volatility models. We consider models of the affine-jump diffusion family and the log-variance specification popular in the econometric literature. We conduct inference within various stochastic volatility models, eventually with jumps, using an efficient adaptive Markov-chain Monte-Carlo procedure, thus generalizing solutions previously proposed in the literature. This methodology effects a sensible reduction in the autocorrelation observed in the Markov chain generated by the volatility-process updating scheme. To rank the competing models, we use the Bayes factor. Because there are many latent components (volatility and jumps), this computation is a challenging task. The posterior distribution at a given point is estimated through a sequence of reduced runs of the MCMC algorithm, which is a particular case of a bridge sampling method. The likelihood is computed using an auxiliary particle filter, which is also used to compute the VaR forecasts to provide further validation of the competing models. We show some results for the Standard & Poor's $500$ index series