Stochastic Volatility Models as Hidden Markov Models and Statistical Applications

Abstract

This paper deals with the fixed sampling interval case for stochastic volatility models. We consider a two-dimensional diffusion process (Y<sub>t</sub>, V<sub>t</sub>), where only (Y<sub>t</sub>) is observed at n discrete times with regular sampling interval ͉. The unobserved coordinate (V<sub>t</sub>) is ergodic and rules the diffusion coefficient (volatility) of (Y<sub>t</sub>). We study the ergodicity and mixing properties of the observations (Y<sub>i͉</sub>). For this purpose, we first present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (V<sub>t</sub>). When the stochastic differential equation of (V<sub>t</sub>) depends on unknown parameters, we derive moment-type estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n<sup>1/2</sup><i/>. Examples of models coming from finance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.