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

TL;DR

This paper develops a rate-optimal estimator for the quadratic variation of a semi-martingale boundary process using high-frequency data, with applications to estimating volatility from limit order book quotes.

Contribution

It introduces a novel estimator based on a Poisson point process framework, connecting to Brownian excursion theory and achieving the optimal convergence rate of n^{-1/3}.

Findings

Achieves the optimal convergence rate of n^{-1/3}.

Connects volatility estimation to Brownian excursion areas.

Provides a potential method for intra-day volatility estimation from order book data.

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.