Stochastic Volatility: A Tale of Co-Jumps, Non-Normality, GMM and High Frequency Data

Abstract

In this article we introduce a linear quadratic volatility model with co-jumps and show how to cal- ibrate this model to a rich dataset. We apply GMM and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both re- turn process and volatility process and the superposition structure of a continuous linear quadratic volatility process and a Lévy-driven Ornstein-Uhlenbeck process. We compare the quality of fit for several models, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normal distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) mo- ment conditions. A Monte Carlo experiments is employed to confirm this.

Keywords: linear quadratic volatility, jump process, general method of moments, power variations, multi-power variations, Monte Carlo

JEL Classification: C51, C52, G17