Inference for Lévy Driven Stochastic Volatility Models Via Adaptive Sequential Monte Carlo

In the following paper we investigate simulation methodology for Bayesian inference in Lévy driven SV models. Typically, Bayesian inference from such statistical models is performed using Markov chain Monte Carlo (MCMC) methods. However, it is well-known that fitting SV models using MCMC is not always straight-forward. One method that can improve over MCMC is SMC samplers ([14]), but in that approach, there can be many user-set parameters that can be difficult to specify. Thus, in this paper, we develop a fully automated sequential Monte Carlo (SMC) algorithm, which substantially improves over the standard Markov chain Monte Carlo (MCMC) methods in the literature. To illustrate our methodology, we look at a model comprised of a Heston type model ([26]) with an independent, additive, variance-Gamma process ([32]) in the returns equation. The infinite activity nature of the driving gamma process can capture the observed behaviour of many financial time series, and a discretized version, fit in a Bayesian manner, has been found to be very useful for modelling equity data ([30]). We demonstrate that it is possible to draw exact inference, in the sense of no time-discretization error, from the Bayesian SV model, by the usage of simple results in stochastic calculus and by an auxiliary variable representation of the posterior. In this case, the model has a transition density that is only known through its characteristic function, but exact inference is still possible.