Quarticity and other functionals of volatility: Efficient estimation

Abstract

We consider a multidimensional Itô semimartingale regularly sampled on $[0,t]$ at high frequency $1/\Delta_{n}$, with $\Delta_{n}$ 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. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most $\Delta_{n}^{1/4}$, this procedure reaches the parametric rate $\Delta_{n}^{1/2}$, as it is usually the case in integrated functionals estimation. After a suitable bias correction, we obtain an unbiased central limit theorem for our estimator and show that it is asymptotically efficient within some classes of sub models.