Random matrix theory of dynamical cross correlations in financial data

Abstract

A new method taking advantage of the random matrix theory is proposed to extract genuine dynamical correlations between price fluctuations of different stocks. One-day returns of 557 Japanese major stocks for the 11-year period from 1996 to 2006 are used for this study. We carry out the discrete Fourier transform of the returns to construct a correlation matrix at each frequency. Also we prepare series of random numbers which are mutually uncorrelated and hence serve as a reference. Comparison of the eigenvalues of the empirical correlation matrix with the reference results of the random one enables us to distinguish between information and noise involved in complicated behavior of the stock returns. It is thus demonstrated that there exist collective motions of the stock prices with periods well over days. Finally we indicate a possible application of the present finding to the risk evaluation of portfolios.