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

Giacomo Toscano, Giulia Livieri, Maria Elvira Mancino and Stefano Marmi

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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.

Date: 2021-12, Revised 2022-09
New Economics Papers: this item is included in nep-ecm and nep-ets
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