A mathematical analysis of the Gumbel test for jumps in stochastic volatility models

This article gives an exhaustive mathematical analysis of the Gumbel test for additive jump components based on extreme value theory. The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view. They consider a continuous-time stochastic volatility model with a general continuous volatility process and observe it under a high-frequency sampling scheme. The 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. Our article presents a moment method based technique that provides some deeper mathematical insights into the convergence and divergence case of the test statistics. In the non-jump case we are able to prove the convergence to the Gumbel distribution under greatly weak assumptions: The volatility process has to be merely pathwise Hölder continuous with an arbitrary random Hölder exponent and we have no restrictions concerning an additional drift term. Therefore, for example, we are allowing for long and short-range dependence. In the case of existing additive jumps, we give divergence results in a general semimartingale setting and investigate the speed of divergence depending on the jump activity. As a by-product of our analysis we also deduce an optimal pathwise estimator for the spot volatility process. Moreover, we provide a detailed simulation study that compares the power of the Gumbel test with the power of the jump test proposed by Barndorff–Nielsen and Shephard in 2006 for Hölder exponents close to zero. Finally, both tests are applied to a real dataset.