Improved volatility estimation based on limit order books

Abstract

For a semi-martingale Xt, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation hX;Xit is con- structed based on observations in the vicinity of Xt. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion ar- eas. A major application is the estimation of the integrated squared volatility of an efficient price process Xt 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 classi- cal n????1=4-rate obtained from transaction prices under microstructure noise.