Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series

Abstract

We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. We find that C has the universal properties of the Gaussian orthogonal ensemble of random matrices. Furthermore, we analyze the eigenvectors of C through their inverse participation ratio and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum-a situation reminiscent of results in localization theory.

Self-organized complexity in economics and finance

“…This large number of elements (1 million) does not frighten a physicist with a computer. Eugene Wigner applied random matrix theory 50 years ago to interpret the complex spectrum of energy levels in nuclear physics (36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48). We do exactly the same thing and apply random matrix theory to the matrix C. We find that certain eigenvalues of that 1,000 ϫ 1,000 matrix deviate from the predictions of random matrix theory, which has not eigenvalues greater than an upper bound of Ϸ2.0.…”

A microscopic model of triangular arbitrage

“…Analyzing correlation in financial time series is a topic of considerable interest [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. We recently pointed out [19,20,21] that, in the foreign exchange market, a correlation among exchange rates can be generated by triangular arbitrage transactions.…”

Towards identifying the world stock market cross-correlations: DAX versus Dow Jones

“…Noise is ubiquitous and overwhelming, and therefore it seems natural that majority of eigenvalues of the stock market correlation matrix agree very well [1,2] with the universal predictions of random matrix theory [3]. This perhaps can be traced back to similar characteristics observed already on the level of human's brain activity [4] Collectivity on the other hand is much more subtle but it is this component which is of principal interest because it accounts for system-specific nonrandom properties and thus potentially encodes the system's future states.…”