Estimating time-changes in noisy Lévy models

In quantitative finance, we often wish to recover the volatility of asset prices given by a noisy It\=o semimartingale. Existing estimates, however, lose accuracy when the jumps are of infinite variation, as is suggested by empirical evidence. In this paper, we show that when the efficient prices are given by an unknown time-changed L\'evy process, the rate of time change, which plays the role of the volatility, can be estimated well under arbitrary jump activity. We further show that our estimate remains valid for the volatility in the general semimartingale model, obtaining convergence rates as good as any previously implied in the literature.