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

Friedrich Hubalek; Petra Posedel Šimović

Vol. 30 No. 1 (2025)

Regular Articles

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

Published 2025-03-11

Friedrich Hubalek
Petra Posedel Šimović

Friedrich Hubalek
TU Wien (Vienna University of Technology)

Petra Posedel Šimović
University of Zagreb, Faculty of Agriculture

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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 the class of non-Gaussian Ornstein-Uhlenbeck based models, as introduced by Barndorff-Nielsen and Shephard. We develop in detail the 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 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 do not use %any ergodicity arguments, but employ a law of large numbers by %Rajchman and a multivariate central limit theorem for martingales. As an illustration we provide a simulation study for daily increments, but the method applies unchanged for 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 estimator and investigate the improvement in reducing variance even though this improvement is not elevate. This paper complements the earlier works \cite{HP2011,HP2013}.

Keywords

Optimal martingale estimating functions, stochastic volatility models with jumps, consistency and asymptotic normality

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Abstract

Keywords

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Author Biography

Friedrich Hubalek

Petra Posedel Šimović