Efficient estimation of integrated volatility in presence of infinite variation jumps

TL;DR

This paper introduces new nonparametric estimators for integrated volatility that remain efficient even with infinite variation jumps, using characteristic functions and bias correction techniques.

Contribution

It develops a novel two-step estimation method that achieves optimal rate and variance in the presence of infinite variation jumps, improving volatility estimation accuracy.

Findings

Estimators achieve the optimal rate of convergence.

Bias correction leads to efficiency and a feasible CLT.

Method handles stochastic stable Lévy process jumps.

Abstract

We propose new nonparametric estimators of the integrated volatility of an It\^{o} semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" L\'{e}vy processes, that is, processes whose L\'{e}vy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.