Stochastic volatility models with possible extremal clustering

Abstract

In this paper we consider a heavy-tailed stochastic volatility model, $X_t=\sigma_tZ_t$, $t\in\mathbb{Z}$, 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.