Analysis of a Class of Likelihood Based Continuous Time Stochastic Volatility Models including Ornstein-Uhlenbeck Models in Financial Economics

Abstract

In a series of recent papers Barndorff-Nielsen and Shephard introduce an attractive class of continuous time stochastic volatility models for financial assets where the volatility processes are functions of positive Ornstein-Uhlenbeck(OU) processes. This models are known to be substantially more flexible than Gaussian based models. One current problem of this approach is the unavailability of a tractable exact analysis of likelihood based stochastic volatility models for the returns of log prices of stocks. With this point in mind, the likelihood models of Barndorff-Nielsen and Shephard are viewed as members of a much larger class of models. That is likelihoods based on n conditionally independent Normal random variables whose mean and variance are representable as linear functionals of a common unobserved Poisson random measure. The analysis of these models is facilitated by applying the methods in James (2005, 2002), in particular an Esscher type transform of Poisson random measures; in conjunction with a special case of the Weber-Sonine formula. It is shown that the marginal likelihood may be expressed in terms of a multidimensional Fourier-cosine transform. This yields tractable forms of the likelihood and also allows a full Bayesian posterior analysis of the integrated volatility process. A general formula for the posterior density of the log price given the observed data is derived, which could potentially have applications to option pricing. We extend the models to include leverage effects in section 5. It is shown that inference does not necessarily require simulation of random measures. Rather, classical numerical integration can be used in the most general cases.