Estimation of volatility functionals: the case of a square root n window
Abstract
We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency 1/n, with n 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/n and minimal asymptotic variance. To achieve this we use spot volatility estimators based on observations within time intervals of length knn. In [5] this was done with kn tending to infinity and knn tending to 0, and a central limit theorem was given after suitable de-biasing. Here we do the same with kn of order 1/n. This results in a smaller bias, although more difficult to eliminate.
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