Asymptotic normality and finite-sample robustness of the Fourier spot volatility estimator in the presence of microstructure noise

We study the efficiency and robustness of the Fourier spot volatility estimator by Malliavin and Mancino (2002) when high-frequency prices are contaminated by microstructure noise. Firstly, we show that the estimator is consistent and asymptotically efficient in the presence of additive noise, establishing a Central Limit Theorem (CLT) with the optimal rate of convergence $n^{1/8}$. Additionally, we complete the asymptotic theory proved by Mancino and Recchioni (2015) in the absence of noise, obtaining a CLT with the optimal rate of convergence $n^{1/4}$. Feasible CLTs with the optimal convergence rate are also obtained, by proving the consistency of the Fourier estimators of the spot volatility of volatility and the quarticity in the presence of noise. Secondly, we introduce a feasible method for selecting the cutting frequencies of the estimator in the presence of noise, based on the optimization of the integrated asymptotic variance. Finally, we provide support to the accuracy and robustness of the method by means of a numerical study and an empirical exercise, which is conducted using tick-by-tick prices of three US stocks with different liquidity.