Some properties of stochastic volatility model that are induced by its volatility sequence

Abstract

In this paper, we consider a heavy-tailed stochastic volatility model $X_t={\sigma }_t{Z}_t$, $t\in Z$, where the volatility sequence  $\left({\sigma }_t\right)$ and the iid noise sequence  $\left(Z_t\right)$ are assumed to be independent, $\left({\sigma }_t\right)$ is regularly varying with index $\alpha >0$, and the $Z_t$’s to have moments of order less than $\alpha /2$. Here, we prove that, under certain conditions, the stochastic volatility model inherits the anti-clustering condition of $\left(X_t\right)$ from the volatility sequence  $\left({\sigma }_t\right)$. Next, we consider a stochastic volatility model in which $\left({\sigma }_t\right)$ is an exponential AR(2) process with regularly varying marginals and show that this model satisfies the regular variation, mixing and anti-clustering conditions in Davis and Hsing (1995).