Spot volatility and covariance estimation on high frequency data using the Fourier methodology: microstructure noise, Limit Order Book and positive semi-definiteness

This work aims at further developing the literature on the Fourier estimator, originally proposed by Malliavin and Mancino (2002), analyzing in particular the spot estimator with high-frequency data which are contaminated by microstructure noise, an aspect that was only partially addressed by previous works. The advances that this work presents with respect to the existing literature are related to three outstanding issues, involving both theoretical and practical aspects of spot volatility estimation. The first point considered is the need, from a theoretical point of view, of providing a Central Limit Theorem in presence of noise for the Fourier spot estimator, thus obtaining a convergence rate and an asymptotic error for the estimator under the presence of additive noise; we also provide optimal convergence rate for the estimator in absence of noise. A side quest related to the findings of the above mentioned CLT is to establish a feasible procedure to optimize the parameters involved in the estimator itself, which is accomplished minimizing the conditional asymptotic error, together with the assessment of its performances. As second point we stress the relevance, in presence of a vast literature on volatility estimation, of a broad analysis of the finite sample performance of integrated and spot volatility estimators, with performances evaluated using a Limit Order Book simulator that we show to be able to better reproduce the microstructure of financial markets with respect to the one previously used in the literature. The impact of the use of different estimators is also evaluated in an exercise of optimal execution, and the Fourier spot estimator is shown to provide a satisfactory performance. Finally, we also underline the interest of considering a multivariate setting for spot volatility estimation in presence of irregularly spaced and asynchronous observations. We present a new Fourier-type estimator which, to the best of our knowledge, is the first one to be proved to provide positive semi-definite estimations, we study the sensitivity to its parameters and we compare its performance with the methods previously proposed in the literature, considering a variety of scenarios both for the noise and the efficient price process. As last contribution, we present a technique able to reduce the computational cost of the positive estimator itself.