The Gumbel test for jumps in stochastic volatility models

Abstract

In this paper we develop a test for jumps based on extreme value theory.We consider a continuous- time stochastic volatility model with a general continuous volatility process, allowing for long- and short-range dependence and observe it under a high-frequency sampling scheme. We show that a certain test statistics based on the maximum of increments converges to the Gumbel distribution under the null hypothesis of no additive jump component and to infinity otherwise. In contrast to most other tests based on power variation our test naturally allows to distinguish between positive and negative jumps. As a by-product of our analysis we also deduce an optimal pathwise estimator for the spot volatility process. In addition we provide a small simulation study and show that our test is more sensitive to jumps with a larger power than the Barndorff-Nielsen and Shephard test based on bipower variation. Finally we apply our results to a real data set of the world stock index.