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

In this paper, we consider a heavy-tailed stochastic volatility model Xt=σtZtXt=σtZt, t∈Zt∈Z, where the volatility sequence  (σt)(σt) and the iid noise sequence  (Zt)(Zt) are assumed to be independent, (σt)(σt) is regularly varying with index α>0, and the ZtZt’s to have moments of order less than α/2α/2. Here, we prove that, under certain conditions, the stochastic volatility model inherits the anti-clustering condition of (Xt)(Xt) from the volatility sequence  (σt)(σt). Next, we consider a stochastic volatility model in which (σt)(σt) 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).