Nonparametric inference for spot volatility in pure-jump semimartingales

TL;DR

This paper develops nonparametric methods for estimating spot volatility in pure-jump semimartingales, analyzing two asymptotic regimes and providing valid inference procedures with practical advantages demonstrated through simulations.

Contribution

It introduces a comprehensive framework for nonparametric spot volatility inference under different asymptotic regimes, including cases with jump activity estimation.

Findings

Fixed-$k$ asymptotics yield better finite-sample accuracy.

Derived nonstandard, non-Gaussian limit distributions for inference.

Valid inference procedures are established for various jump activity scenarios.

Abstract

We provide a comprehensive analysis of spot volatility inference in pure-jump semimartingales under two asymptotic settings: fixed-$k$, where each local window uses a fixed number of observations, and large-$k$, where this number grows with sampling frequency. For both active- and possibly inactive-jump settings, we derive generally nonstandard, typically non-Gaussian limit distributions and establish valid inference, including when the jump-activity index is consistently estimated. Simulations show that fixed-$k$ asymptotics offer markedly better finite-sample accuracy, underscoring their practical advantage for nonparametric spot volatility inference.