Volatility of volatility estimation: central limit theorems for the Fourier transform estimator and empirical study of the daily time series stylized facts

TL;DR

This paper develops and analyzes Fourier-based estimators for the integrated volatility of volatility, establishing their asymptotic properties, and applies them to long-term financial data to uncover stylized facts about volatility dynamics.

Contribution

It introduces bias-corrected and non-bias-corrected Fourier estimators for volatility of volatility and proves their asymptotic normality, with empirical validation and application to market data.

Findings

Bias-corrected estimator achieves optimal $n^{1/4}$ convergence rate.

Simulation results confirm theoretical asymptotic distribution.

Reconstruction of daily volatility of volatility reveals stylized facts.

Abstract

We study the asymptotic normality of two feasible estimators of the integrated volatility of volatility based on the Fourier methodology, which does not require the pre-estimation of the spot volatility. We show that the bias-corrected estimator reaches the optimal rate $n^{1/4}$, while the estimator without bias-correction has a slower convergence rate and a smaller asymptotic variance. Additionally, we provide simulation results that support the theoretical asymptotic distribution of the rate-efficient estimator and show the accuracy of the latter in comparison with a rate-optimal estimator based on the pre-estimation of the spot volatility. Finally, using the rate-optimal Fourier estimator, we reconstruct the time series of the daily volatility of volatility of the S\&P500 and EUROSTOXX50 indices over long samples and provide novel insight into the existence of stylized facts about the volatility of volatility dynamics.