Asymptotic analysis for an optimal estimating function for Barndorff-Nielsen Shephard stochastic volatility models

Abstract

We provide and analyze optimal estimators from a fixed sample and asymptotic point of view for a class of discretely observed continuous-time stochastic volatility models with jumps. In particular, we consider a class of non-Gaussian Ornstein-Uhlenbeck-based models, as introduced by Barndorff-Nielsen and Shephard. We develop in detail a martingale estimating function approach for this kind of processes, which are bivariate Markov processes that are not diffusions, but admit jumps. We assume that the bivariate process is observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We prove rigorously consistency and asymptotic normality of the optimal estimator based on a single assumption that all moments of the stationary distribution of the variance process are finite, and give explicit expressions for the asymptotic covariance matrix. As an illustration, we provide a simulation study for daily increments, but the method applies unchanged to any time-scale, including high-frequency observations, without introducing any discretization error. Additionally, we compare the asymptotic covariance matrix of the optimal estimator with the one of the simple explicit estimators and investigate the improvement in variance reduction, even though this improvement is not significant.