No abstract

We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues lambda(i) of C against a "null hypothesis"--a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [lambda(-),lambda(+)] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these "deviating eigenvectors" are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return.

The understanding of complex systems has become a central issue because such systems exist in a wide range of scientific disciplines. We here focus on financial markets as an example of a complex system. In particular we analyze financial data from the S&P 500 stocks in the 19-year period 1992–2010. We propose a definition of state for a financial market and use it to identify points of drastic change in the correlation structure. These points are mapped to occurrences of financial crises. We find that a wide variety of characteristic correlation structure patterns exist in the observation time window, and that these characteristic correlation structure patterns can be classified into several typical “market states”. Using this classification we recognize transitions between different market states. A similarity measure we develop thus affords means of understanding changes in states and of recognizing developments not previously seen.

A new derivation of Dyson’s k-level correlation functions of the Gaussian unitary ensemble (GUE) is given. The method uses matrices with graded symmetry. The number of integrations needed for the ensemble average becomes independent of the level number N. For arbitrary level number N, the k-level correlation function is expressed as an integral involving the eigenvalues of a 2k×2k graded matrix. The limit of infinitely many levels N→∞ is calculated by a simple saddle-point approximation of this integral, avoiding the introduction of Hermite polynomials and oscillator wave functions.