Before revision:  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.
After revision:  In this paper, we show that when the efficient prices are given by an unknown time-changed Levy process; this model is both identifiable, and replicates salient features of price data. We give a new estimate of the rate process in this model, which governs its volatility. Our estimate obtains minimax convergence rates, and is unaffected by arbitrary jump activity.