Fourier series method for measurement of multivariate volatilities
This article deals with a new method based on Fourier analysis to measure volatility under the hypothesis that all observed data \(u^{*}\) are driven by semi-martingales which have Itô's stochastic differential \(du^{j}=\sum_{i=1}^{d}\alpha_{i}^{j} dx^{i}+\beta^{j} dt\), where \(x^{*}\) are independent Brownian motions and \(\alpha_{*}^{*}, \beta^{*}\) are functions depending on time. The matrix \(\sum^{j,k}(t)=\sum_{i=1}^{d}\alpha_{i}^{j}(t)\alpha_{i}^{k}(t)\) is called volatility matrix. The proposed algorithm reconstructs the volatility as a function of time. This method, based on the integration of the time series, is more robust than the classical one, based on the quadratic variation formula, and is well suited to compute volatility for high frequency time series.
scientific article; zbMATH DE number 1833867