Stochastic dynamical model for stock-stock correlations

Wen-Jong Ma, Chin-Kun Hu, and Ravindra E. Amritkar

Phys. Rev. E 70, 026101 – Published 4 August 2004

Abstract

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.

Received 2 October 2003

DOI:https://doi.org/10.1103/PhysRevE.70.026101

Authors & Affiliations

Wen-Jong Ma and Chin-Kun Hu^*

Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan

Ravindra E. Amritkar

Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India

^*Electronic address: huck@phys.sinica.edu.tw

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Vol. 70, Iss. 2 — August 2004

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Images

Figure 1
(Color) The simulated eigenvalue distribution $ρ (λ)$ with $ϵ_{g} \neq 0$ (thick solid line) and $ϵ_{g} = 0$ (dashed line) with $N = 345$ stocks and $T = 3250$ steps, in comparison with the market data (open and filled circles, see text) and the prediction of the random matrix theory [13] (dotted line). The inset shows large isolated eigenvalues, including $λ_{M}$ . The fit (thick solid line) in the figure is obtained with $ϵ_{M} = 0.794$ , 8 major groups, each occupied by more than 10 stocks, and 17 small groups. The number of stocks $n_{g}$ in each group and the corresponding couplings $ϵ_{g}$ are listed in Table . The simulated data are obtained by averaging over an ensemble of 106 matrices.Reuse & Permissions
Figure 2
(Color) The plot of the ensemble averaged value ${\bar{λ}}_{M}$ and the width $Δ λ_{M}$ of the distribution of $λ_{M}$ in the ensemble corresponding to the market mode for various values of $ϵ_{M} (ϵ_{g} = 0.0)$ for three different systems with $N = (345, 406, 1000)$ walks, lengths $T = (3250, 1309, 6448)$ , and over ensembles of $(106, 400, 50)$ cross correlation matrices, respectively. [These are the same $N$ and $T$ values as the pools of stocks analyzed in this study (Fig. 1), Refs. 3 and [4], respectively.] The corresponding curves for the analytical expression, Eq. (15), for finite $N$ (thin broken, dotted, and dashed lines are for $N = 345$ , 406, and 1000, respectively) are plotted for comparison. The series of curve approaches in increasing $N$ to that of Eq. (16) (solid line), which is a power law with exponent 2 for $ϵ_{M} \approx 1$ . The inset shows the same data in log-log plot.Reuse & Permissions
Figure 3
The components (scaled fluctuations of returns) for the eigenvectors of a few typical eigenvalues corresponding to our model (a) and to the market data (b) as described in Fig. 1. They are the eigenvectors of (top down) the market mode $λ_{M}$ , the next three largest eigenvalues $(λ_{1}, λ_{2}, λ_{3})$ and two typical modes with their eigenvalues falling inside the bulk $(λ_{4})$ and at the lower edge of the bulk component $(λ_{5})$ , respectively. The stocks are numbered so that the stocks belonging to the same group are consecutively placed and the groups are arranged in the plots, from left to right, according to their sizes in the descending order. There are 27 stocks placed at the end that do not belong to any group. The vertical lines indicate the boundaries of the groups and the horizontal lines mark the zero levels.Reuse & Permissions
Figure 4
(Color) The 345 US stocks classified by our method and by the minimum spanning tree analysis on the correlation coefficients in Ref. 14. We label the stocks in the eight largest groups obtained by our method (see text and Table ) in different colors. The stocks are also marked by different symbols according to their sectors in S&P 500 classification. In our approach, the grouping is realized as the perturbing part of the cooperative activities in the fluctuations to the dominant market activities (see text). In the minimum spanning tree analysis, there is no such division.Reuse & Permissions

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