Volatility estimation under one-sided errors with applications to limit order books

Abstract

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. We derive $n^{-1/3}$ as optimal convergence rate in a high-frequency framework with $n$ observations (in mean). We discuss a potential application for the estimation of the integrated squared volatility of an efficient price process $X_{t}$ from intra-day order book quotes.