A stochastic volatility model with flexible extremal dependence structure

Abstract

Stochastic volatility processes with heavy-tailed innovations are a well-known model for financial time series. In these models, the extremes of the log returns are mainly driven by the extremes of the i.i.d. innovation sequence which leads to a very strong form of asymptotic independence, that is, the coefficient of tail dependence is equal to $1/2$ for all positive lags. We propose an alternative class of stochastic volatility models with heavy-tailed volatilities and examine their extreme value behavior. In particular, it is shown that, while lagged extreme observations are typically asymptotically independent, their coefficient of tail dependence can take on any value between $1/2$ (corresponding to exact independence) and 1 (related to asymptotic dependence). Hence, this class allows for a much more flexible extremal dependence between consecutive observations than classical SV models and can thus describe the observed clustering of financial returns more realistically. The extremal dependence structure of lagged observations is analyzed in the framework of regular variation on the cone $(0,\infty)^d$. As two auxiliary results which are of interest on their own we derive a new Breiman-type theorem about regular variation on $(0,\infty)^d$ for products of a random matrix and a regularly varying random vector and a statement about the joint extremal behavior of products of i.i.d. regularly varying random variables.