Stochastic volatility models with possible extremal clustering

Abstract

In this paper we consider a heavy-tailed stochastic volatility model, $X_{t}=\sigma_{t}Z_{t}$, $t\in\mathbb{Z}$, where the volatility sequence $(\sigma_{t})$ and the i.i.d. noise sequence $(Z_{t})$ are assumed independent, $(\sigma_{t})$ is regularly varying with index $\alpha>0$, and the $Z_{t}$’s have moments of order larger than $\alpha$. In the literature (see Ann. Appl. Probab. 8 (1998) 664–675, J. Appl. Probab. 38A (2001) 93–104, In Handbook of Financial Time Series (2009) 355–364 Springer), it is typically assumed that $(\log\sigma_{t})$ is a Gaussian stationary sequence and the $Z_{t}$’s are regularly varying with some index $\alpha$ (i.e., $(\sigma_{t})$ has lighter tails than the $Z_{t}$’s), or that $(Z_{t})$ is i.i.d. centered Gaussian. In these cases, we see that the sequence $(X_{t})$ does not exhibit extremal clustering. In contrast to this situation, under the conditions of this paper, both situations are possible; $(X_{t})$ may or may not have extremal clustering, depending on the clustering behavior of the $\sigma$-sequence.