Estimation of volatility functionals: The case of a √n window

Abstract

We consider a multidimensional Itô semimartingale regularly sampled on [0, t] at high frequency 1/Δn, with Δn going to zero. The goal of this paper is to provide an estimator for the integral over [0, t] of a given function of the volatility matrix, with the optimal rate 1/ √Δn and minimal asymptotic variance. To achieve this, we use spot volatility estimators based on observations within time intervals of length knΔn. In [5], thiswas done with kn →∞and kn √Δn → 0, and a central limit theorem was given after suitable de-biasing. Here we do the same with the choice kn ≍ 1/ √Δn. This results in a smaller bias, although more difficult to eliminate.