Quantifying and interpreting collective behavior in financial markets

Abstract

Firms having similar business activities are correlated. We analyze two different cross-correlation matrices C constructed from (i) 30-min price fluctuations of 1000 US stocks for the two-year period 1994-95 and (ii) one-day price fluctuations of 422 US stocks for the 35-year period 1962-96. We find that the eigenvectors of C corresponding to the largest eigenvalues allow us to partition the set of all stocks into distinct subsets. These subsets are similar to business sectors, and are stable for extended periods of time. We find that price fluctuations of these subsets are characterized by power-law decaying time correlations, reminiscent of strongly interacting systems.

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.…”

“…Furthermore, the content of the eigenvectors corresponding to those eigenvalues correspond to well-defined business sectors. This allows us to define business sectors without knowing anything about the separate stocks: a Martian who cannot understand stock symbols could nonetheless identify business sectors (46,47).…”

Cluster analysis for portfolio optimization

“…Different elements are iteratively included in clusters starting from the first two elements of the similatity measure ordered list. At each step, when two elements or one element and (8) indicating that the similarity between any element of cluster t and any element of cluster r is the similarity between the two most similar entities in clusters t and r. Conversely, if s ij is a distance-like measure…”

“…The results of these investigations have been obtained by using concepts and tools of random matrix theory (RMT) [3]. The RMT quantification of the statistical uncertainty associated with the estimation of the correlation coefficient matrix of a finite multivariate time series has been recently used to device a procedure to filter the information present in the correlation coefficient matrix which is robust with respect to the unavoidable statistical uncertainty (in the econophysics literature it has been used the term of noise dressing) [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. The correlation matrices obtained by this filtering procedure has been used in portfolio optimization.…”

Spatial dependence in stock returns: local normalization and VaR forecasts

“…Several methods have been proposed in the past to reduce the noise, while keeping the times series short, e.g. see Gopikrishnan et al (2001) and Giada and Marsili (2001).…”