In recent years, there has been substantive empirical evidence that
stochastic volatility is rough. In other words, the local behavior of
stochastic volatility is much more irregular than semimartingales and resembles
that of a fractional Brownian motion with Hurst parameter H<0.5. In this
paper, we derive a consistent and asymptotically mixed normal estimator of H
based on high-frequency price observations. In contrast to previous works, we
work in a semiparametric setting and do not assume any a priori relationship
between volatility estimators and true volatility. Furthermore, our estimator
attains a rate of convergence that is known to be optimal in a minimax sense in
parametric rough volatility models