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Volatility estimation under one-sided errors with applications to 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. We derive n1/3n^{-1/3} as optimal convergence rate in a high-frequency framework with nn observations (in mean). We discuss a potential application for the estimation of the integrated squared volatility of an efficient price process XtX_t from intra-day order book quotes.

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