Improved volatility estimation based on limit order books

For a semi-martingale $X_t$, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation $\langle X, X \rangle_t$ is constructed based on observations in the vicinity of $X_t$. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. A major application is the estimation of the integrated squared volatility of an efficient price process $X_t$ from intra-day order book quotes. We derive $n^{-1/3}$ as optimal convergence rate of integrated squared volatility estimation in a high-frequency framework with $n$ observations (in mean). This considerably improves upon the classical $n^{-1/4}$-rate obtained from transaction prices under microstructure noise. %An estimator based on local order statistics attaining the rate is presented.