Estimation of volatility functionals: the case of a square root n window

TL;DR

This paper develops an estimator for integrated volatility functionals of multidimensional Ito semimartingales, achieving optimal convergence rates and minimal asymptotic variance using high-frequency data and a novel window size choice.

Contribution

It introduces a new method for estimating volatility functionals with a window size of order 1/\u221a{ ext{ extbackslash}Delta ext{ extbackslash}n}, improving bias properties and asymptotic efficiency over previous approaches.

Findings

Achieves the optimal rate of 1/ ext{ extbackslash}sqrt{ ext{ extbackslash}Delta ext{ extbackslash}n} for the estimator.

Provides a central limit theorem for the proposed estimator.

Demonstrates reduced bias with the new window size choice.

Abstract

We consider a multidimensional Ito 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, with the optimal rate 1/\sqrt{\Delta_n} and minimal asymptotic variance. To achieve this we use spot volatility estimators based on observations within time intervals of length k_n\Delta_n. In [5] this was done with k_n tending to infinity and k_n\sqrt{\Delta_n} tending to 0, and a central limit theorem was given after suitable de-biasing. Here we do the same with k_n of order 1/\sqrt{\Delta_n}. This results in a smaller bias, although more difficult to eliminate.