Efficient Integrated Volatility Estimation in the Presence of Infinite Variation Jumps via Debiased Truncated Realized Variations

TL;DR

This paper introduces a new high-order, rate- and variance-efficient estimator for the integrated volatility of Itô semimartingales with jumps of unbounded variation, using a two-step debiasing method based on truncated realized quadratic variation.

Contribution

It develops a novel estimator that handles jumps of unbounded variation by leveraging high-order expansions and a two-step debiasing process, extending existing methods.

Findings

Outperforms existing estimators in simulations.

Effective for jumps with Blumenthal-Getoor index in (1,8/5).

Works also for jumps with index less than 1.

Abstract

Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than two decades. One of the most well-known and widely studied problems has been the estimation of the quadratic variation of the continuous component of an It\^o semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed in the literature when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of the truncated moments of a locally stable L\'evy process, we propose a new rate- and variance-efficient volatility estimator for a class of It\^o semimartingales whose jumps behave locally like those of a stable L\'evy process with Blumenthal-Getoor index $Y\in (1,8/5)$ (hence, of unbounded variation). The proposed method is based on a two-step debiasing procedure for the truncated realized quadratic variation of the process and can also cover the case $Y<1$. Our Monte Carlo experiments indicate that the method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.