Efficient Volatility Estimation for L\'evy Processes with Jumps of Unbounded Variation

TL;DR

This paper introduces a new high-order estimator for the quadratic variation of Le9vy processes with unbounded jumps, outperforming existing methods in high-frequency financial data analysis.

Contribution

It develops a novel rate- and variance-efficient estimator for Le9vy processes with unbounded variation, using a two-step debiasing approach based on high-order expansions.

Findings

The estimator outperforms existing methods in simulations.

It effectively handles jumps of unbounded variation.

The method is applicable to processes with stable-like small jumps.

Abstract

Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than a decade. One of the most well-known and widely studied problems is that of 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 L\'evy process, we construct a new rate- and variance-efficient estimator for a class of L\'evy processes of unbounded variation, whose small jumps behave like those of a stable L\'evy process with Blumenthal-Getoor index less than $8/5$. The proposed method is based on a two-step debiasing procedure for the truncated realized quadratic variation of the process. Our Monte Carlo experiments indicate that the method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.