Estimating covariance via Fourier methodin the presence of asynchronous trading and microstructure noise.

We analyze the effects of market microstructure noise on the Fourier estimator of multivariate volatilities. We prove that the estimator is consistent in the case of asynchronous data and asymptotically unbiased in the presence of various types of microstructure noise. This result is obtained through an analytical computation of the bias and the mean squared error of the Fourier estimator and confirmed by Monte Carlo experiments. A comparison with several covariance estimators is performed.