  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.
