Random Matrix Theory: applications to financial data analysis

Random Matrix Theory (RMT) was initially devised in the early fifties to study the physical properties of heavy nuclei. After its early successes in Nuclear Physics, RMT found more and more applications in very diverse fields, ranging from several areas of theoretical Physics to Pure Mathematics, from Genomics to Information Theory. In this talk, I will first review the main applications of RMT to the analysis of financial data, with particular emphasis on the analysis of the spectral properties of financial correlation matrices. Empirically observed eigenvalue spectra typically display a complex structure made of a main bulk around 1 (for standardized data) plus a small number of larger eigenvalues "leaking out" of the bulk. From the viewpoint of RMT, the main eigenvalue bulks of empirical spectra have been identified, in a number of works, with the Marcenko-Pastur distribution, i.e. the average spectral density of random Gaussian correlation matrices. Thus, according to this picture, the main part of empirical spectra can essentially be regarded as noise, whereas the larger eigenvalues out of the bulk are interpreted as the only ones yielding genuine information on the correlation structure of the set of stocks under study. In the work to be presented, the common knowledge described above is challenged and critically revised. In particular, I will show how the empirically observed eigenvalue bulks emerge as superpositions of smaller structures, which in turn emerge due to cross-correlations within clusters of stocks. This will be shown by acting on financial correlation matrices with a suitable filtering technique, which allows to perform significant comparisons between empirically observed facts and RMT predictions for the correlation matrices of factor models. A good qualitative agreement between data and theory is found, and the discrepancies between the two are shown to be due to further unresolved correlation structures.