Asymptotic Normality for the Fourier spot volatility estimator in the presence of microstructure noise

TL;DR

This paper proves the asymptotic normality and efficiency of the Fourier spot volatility estimator in noisy high-frequency data, introducing a new adaptive parameter selection method and extending existing theory.

Contribution

It establishes the asymptotic properties of the Fourier spot volatility estimator under microstructure noise and proposes a novel adaptive method for parameter selection.

Findings

Proves a CLT with rate n^{1/8} in noisy settings.

Extends asymptotic theory to noise-free case with rate n^{1/4}.

Introduces a feasible adaptive parameter selection method.

Abstract

The main contribution of the paper is proving that the Fourier spot volatility estimator introduced in [Malliavin and Mancino, 2002] is consistent and asymptotically efficient if the price process is contaminated by microstructure noise. Specifically, in the presence of additive microstructure noise we prove a Central Limit Theorem with the optimal rate of convergence $n^{1/8}$. The result is obtained without the need for any manipulation of the original data or bias correction. Moreover, we complete the asymptotic theory for the Fourier spot volatility estimator in the absence of noise, originally presented in [Mancino and Recchioni, 2015], by deriving a Central Limit Theorem with the optimal convergence rate $n^{1/4}$. Finally, we propose a novel feasible adaptive method for the optimal selection of the parameters involved in the implementation of the Fourier spot volatility estimator with noisy high-frequency data and provide support to its accuracy both numerically and empirically.