Fourier series method for measurement of multivariate volatilities

We present a methodology based on Fourier series analysis to compute time series volatility when the data are observations of a semimartingale. The procedure is not based on the Wiener theorem for the quadratic variation, but on the computation of the Fourier coefficients of the process and therefore it relies on the integration of the time series rather than on its differentiation. The method is fully model free and nonparametric. These features make the method well suited for financial market applications, and in particular for the analysis of high frequency time series and for the computation of cross volatilities.