Applying Free Random Variables to Random Matrix Analysis of Financial Data. Part I: A Gaussian Case

We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing for example in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of the free random variables calculus, with a view to promote them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as of an universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.