Testing for Rough Volatility When Prices Are Purely Discontinuous

ABSTRACT We consider the problem of nonparametric testing for rough volatility, using high‐frequency data with a fixed time span, in a setting where the price is purely discontinuous. More specifically, we analyze the asymptotic properties of a test we developed in previous work in a pure‐jump setting. We show that the asymptotic behavior of the test remains unchanged, regardless of whether the underlying price contains a diffusion or is of purely discontinuous type. This adaptive nature of the test is very convenient from an empirical point of view as it implies that no pretest for the underlying features of the price process, that is, whether it contains a diffusion, is needed when testing for rough volatility. A Monte Carlo study shows good finite sample behavior of the test in purely discontinuous price settings.