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

摘要

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