Glasnik Matematicki, Vol. 48, No. 1 (2013), 185-210.

ASYMPTOTIC ANALYSIS AND EXPLICIT ESTIMATION OF A CLASS OF STOCHASTIC VOLATILITY MODELS WITH JUMPS USING THE MARTINGALE ESTIMATING FUNCTION APPROACH

Friedrich Hubalek and Petra Posedel

Vienna University of Technology, Financial and Actuarial Mathematics, Wiedner Hauptstraße 8 / 105-1, A-1040 Vienna, Austria
e-mail: fhubalek@fam.tuwien.ac.at

Zagreb School of Economics and Management, Jordanovac 100, 10000 Zagreb, Croatia
e-mail: pposedel@zsem.hr

Abstract.   We provide and analyze explicit estimators 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 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. 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.

2010 Mathematics Subject Classification.   60G51, 62F12, 62M05.

Key words and phrases.   Martingale estimating functions, stochastic volatility models with jumps, consistency and asymptotic normality.

Full text (PDF) (free access)

DOI: 10.3336/gm.48.1.15

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