Mixed semimartingales: Volatility estimation in the presence of rough noise

We consider the problem of estimating volatility based on high-frequency data when the observed price process is a continuous It\^o semimartingale contaminated by microstructure noise. Assuming that the noise process is compatible across different sampling frequencies, we argue that it typically has a similar local behavior to fractional Brownian motion. For the resulting class of processes, which we call mixed semimartingales, we derive consistent estimators and asymptotic confidence intervals for the roughness parameter of the noise and the integrated price and noise volatilities, in all cases where these quantities are identifiable. Our model can explain key features of recent stock price data, most notably divergence rates in volatility signature plots that vary considerably over time and between assets.