The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price process

TL;DR

This paper analyzes the Fourier estimator for spot volatility, demonstrating its convergence under unbounded volatility and jump processes, and establishing new rates and almost sure convergence results.

Contribution

It extends the Fourier estimator's theoretical framework to include unbounded volatility and jump processes, providing convergence rates and almost sure convergence proofs.

Findings

Convergence of the Fourier estimator is established for unbounded volatility.

The estimator converges almost surely, not just in probability.

Rescaled polynomial approximates the quadratic jump process.

Abstract

In this paper we study the Fourier estimator of Malliavin and Mancino for the spot volatility. We establish the convergence of the trigonometric polynomial to the volatility's path in a setting that includes the following aspects. First, the volatility is required to satisfy a mild integrability condition, but otherwise allowed to be unbounded. Second, the price process is assumed to have cadlag paths, not necessarily continuous. We obtain convergence rates for the probability of a bad approximation in estimated coefficients, with a speed that allow to obtain an almost sure convergence and not just in probability in the estimated reconstruction of the volatility's path. This is a new result even in the setting of continuous paths. We prove that a rescaled trigonometric polynomial approximate the quadratic jump process.