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Improved volatility estimation based on limit order books

16 August 2014
M. Bibinger
M. Jirak
M. Reiß
ArXiv (abs)PDFHTML
Abstract

For a semi-martingale XtX_tXt​, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation ⟨X,X⟩t\langle X, X \rangle_t⟨X,X⟩t​ is constructed based on observations in the vicinity of XtX_tXt​. 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_tXt​ from intra-day order book quotes. We derive n−1/3n^{-1/3}n−1/3 as optimal convergence rate of integrated squared volatility estimation in a high-frequency framework with nnn observations (in mean). This considerably improves upon the classical n−1/4n^{-1/4}n−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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