We propose a model of coupled random walks for stock-stock correlations. The walks in the model are coupled via a mechanism that the displacement (price change) of each walk (stock) is activated by the price gradients over some underlying network. We assume that the network has two underlying structures, describing the correlations among the stocks of the whole market and among those within individual groups, respectively, each with a coupling parameter controlling the degree of correlation. The model provides the interpretation of the features displayed in the distribution of the eigenvalues for the correlation matrix of real market on the level of time sequences. We verify that such modeling indeed gives good fitting for the market data of US stocks. In recent decades, the analysis of many physical and social systems has been based on the idea that randomness in the fluctuations is hallmarked by certain prototypes, such as eigenvalue distribution of a random matrix, and any deviation from the latter is a result of the presence of correlations. Such an idea has been used to study the level statistics [1] of locally activated states [2] for electrons in heavy atoms or in solids, atomic vibrations in spatially disordered systems, distribution spectra of eigenvalues for the correlation matrices of stocks [3,4], internet traffic [5], and atmospheric fluctuations [6]. While recent advances [3-6] suggest a robust approach to reveal cross correlations from the data of time sequences, a comprehensive address of the shared dynamics behind these different systems, however, is still lacking. In the present paper, we will address this problem by proposing a stochastic dynamic model for stock-stock correlations [3,4]. Our approach is useful for studying time evolution of other interacting many-body systems subject to random noises. The nature of fluctuations in financial markets [3,4,7-11] has been of interest to the traders as well as a variety of professionals for more than a century. If such fluctuations could be completely characterized by random walks, as first proposed by Bachelier in 1900 [9], making profit under controlled risks through the transactions in the markets would seem impractical. Correlations in such fluctuations have been demonstrated in recent studies [3,4] of the distribution spectrum of eigenvalues of the cross correlation matrix for the price changes of stocks in real markets. The matrix measures the statistical overlap of the fluctuations [3,4,11,12] in the price changes (the returns) between pairs of stocks. The spectrum from market data [3,4] possesses a bulk of continuously distributed eigenvalues, which is similar to the prototype predicted by the level statistics of the random matrix theory (RMT) [1,4,13] and may be considered as mainly contributed by the randomness. The effects of the correlations [3,4,11] manifest in the eigenvectors of those eigenvalues isolated from the bulk, which include one large eigenvalue M , corresponding to the correlation among all stocks (market mode) [4] and several much smaller ones scattered in between M and the bulk component. The patterns in the eigenvectors of these latter eigenvalues were related to the presence of groups of correlated stocks [4,11]. In addition, the bulk component also shows some important deviations from RMT predictions. The spectrum of eigenvalues and the corresponding eigenvectors are related to the cooperative behavior in the fluctuations of the stock prices, which is not visible in the local information of the individual correlation coefficients composing that matrix [14]. Based on former information, the connections between the deviated eigenvalues and the presence of correlated groups of stocks have been clarified [4] and rewarded with an ansatz [11] which was applied to modeling the real market data [15]. In the model [11,15], the return of each stock is linearly decomposed into two uncorrelated fluctuating parts. The interdependence of stocks within a group is carried by the part which is synchronously shared by all stocks within that group. This formulation refines the conventional multifactor model [8] leading to blocked structures in its correlation matrix [11], which reproduces those spectral features observed in market data [15]. It assumes that each of the isolated eigenvalues from the bulk is contributed by one block of submatrix, with its eigenvector containing only one activated group. This puts a limitation on the model, not able to digest completely the information carried by the eigenvectors for those deviated eigenvalues in the market data, in which the shared activated stocks are present very often among different eigenmodes. Our paper presents a new formulation to include this fact, at the same time, retaining the realization of grouping in its simplest form. Our approach is a general formulation which describes the cooperative activities in the financial fluctuations beyond the statistics of matrices, on the level of time sequences. *Electronic address: huck@phys.sinica.edu.tw PHYSICAL REVIEW E
A model of coupled random walks is proposed in REF to model the stock correlations, and the walks are coupled via the mechanism that the price change of each stock is activated by the price gradients over some underlying network.