Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models

TL;DR

This paper introduces a simple explicit estimator for Barndorff-Nielsen and Shephard stochastic volatility models, establishing its consistency and asymptotic normality under finite moments, and providing explicit covariance expressions.

Contribution

It develops a martingale estimating function approach for a non-diffusive bivariate model with jumps, without relying on ergodicity, and offers a theoretical foundation for future practical and empirical methods.

Findings

Proves estimator consistency and asymptotic normality.

Provides explicit asymptotic covariance matrix.

Analyzes a non-diffusive model with jumps.

Abstract

We provide a simple explicit estimator for discretely observed Barndorff-Nielsen and Shephard models, prove rigorously consistency and asymptotic normality based on the single assumption that all moments of the stationary distribution of the variance process are finite, and give explicit expressions for the asymptotic covariance matrix. We develop in detail the martingale estimating function approach for a bivariate model, that is not a diffusion, but admits jumps. We do not use ergodicity arguments. We assume that both, logarithmic returns and instantaneous variance are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. As the instantaneous variance is not observable in practice, our results cannot be applied immediately. Our purpose is to provide a theoretical analysis as a starting point and benchmark for further developments concerning optimal martingale estimating functions, and for theoretical and empirical investigations, that replace the variance process with a substitute, such as number or volume of trades or implied variance from option data.