Do price and volatility jump together?∗

We consider a process Xt, which is observed on a finite time interval [0, T ], at discrete times 0, ∆n, 2∆n, · · · . This process is an Itô semimartingale with stochastic volatility σ t . 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 Xt. 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 ∆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.