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Improved volatility estimation based on limit order books

Abstract

For a semi-martingale XtX_t, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation X,Xt\langle X, X \rangle_t is constructed based on observations in the vicinity of XtX_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 XtX_t from intra-day order book quotes. We derive n1/3n^{-1/3} as optimal convergence rate of integrated squared volatility estimation in a high-frequency framework with nn observations (in mean). This considerably improves upon the classical n1/4n^{-1/4}-rate obtained from transaction prices under microstructure noise. %An estimator based on local order statistics attaining the rate is presented.

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