Extremal Memory of Stochastic Volatility with an Application to Tail Shape Inference

ABSTRACT We characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of mul-tivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail depen-dence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter ¯nding by simulation. 1