Testing and inference for fixed times of discontinuity in semimartingales

Abstract

We develop a nonparametric test for deciding whether a semimartingale process, modeling an asset price, contains a fixed time of discontinuity, i.e., a positive probability of a jump, at a given point in time, and we further propose a rate-optimal estimator of the jump distribution when this is the case. Itô semimartingales used commonly in applied work have absolutely continuous in time, with respect to Lebesgue measure, jump compensators, and this rules out fixed times of discontinuity in their paths. However, certain phenomena, such as scheduled economic announcements in finance, make the existence of such discontinuities a possibility. The inference in the paper is based on noisy observations of options written on the asset with different strikes and two different expiration dates. The asymptotics is joint in which the times to maturity of the options shrink to zero and the number of observed options increases to infinity. The test is based on estimates of the characteristic function of the increments of the semimartingale, constructed from the option data, and the fact that the asymptotic limit of the increments and their characteristic functions is different with and without fixed time of discontinuity. The limit distribution of the test statistic is derived and feasible inference is developed on the basis of wild bootstrap type techniques. A Monte Carlo and an empirical illustration show the applicability of the developed inference procedures.