Do price and volatility jump together?

Abstract

We consider a process X_t, which is observed on a finite time interval [0, T], at discrete times 0, Δ_n, 2Δ_n, …. This process is an Itô semimartingale with stochastic volatility σ_t². Assuming that X has jumps on [0, T], we derive tests to decide whether the volatility process has jumps occurring simultaneously with the jumps of X_t. There are two different families of tests for the two possible null hypotheses (common jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh Δ_n goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use on S&P 500 index data.