Financial Markets, High-dimensional Cross-correlation Matrix Structure Evolution

As the blockscone of modern investment theory, the research of the equal-time cross correlation matrix (CCM or covariance matrix) of financial time series has a long history. The traditional econometrics mainly focused on how to construct an effective investment portfolio but without any concerning about the basic properties of itself. While constructing the investment portfolio in such research, only a few (always less than 20) financial assets are considered. However, a high-dimension correlation matrix that is computed by a large number of the prices time series in the same market may hide some key information about the structure characteristics of this market. The methodology of the traditional econometrics cannot deal with such problem, which means we have to try some new. In this paper, I try to talk about this problem from the view of complexity science by using the mature theory and methodology in econophysics and complex network to analyze the structure and evolution properties of CCM.After introducing the research progress in econophysics and complex network, I use three methods: the minimal spanning tree, the random matrix theory and community detection to simultaneously survey the CCM of Chinese stock market (CSM) and New York stock market (NYSE), which respectively contains 249 stocks and 259 stocks. The financial time series I collected are from the year 1997 to 2007(8). Firstly, I use the total data to construct the Static Cross-Correlation Matrix (S-CCM) to investigate the long term stable structure properties. Secondly, I split the ten years data into several small parts, each contains 500 prices, and two contiguous parts just have the difference of only 20 prices. Then we use these divided data to construct a series of Dynamics Cross-Correlation Matrix (D-CCM) to investigate the dynamics evolution properties.The empirical result of the three different research methods shows highly consistency: we find the classification by algorithm does not correspond to the standard industry classification in CSM market, but is almost the same in the NYSE market. We give two financial meaning for the differences between these two markets: Firstly, Investors concern different information in the two markets. Chinese investors pay more attention on the performance of listed companies over the last few years while U.S. investors focus on the prospects of the industry of the listed companies. Secondly, the fundamental in the two markets is not the same. The secondary industry represented by Consumer goods and Industry is the core business sector of CSM, while in NYSE is the tertiary industry represented by Finance. In CSM, the correlation reflects the Industry chain relationship between manufacturing company, and the classification by community detection doesn't correspond to the standard industry classification. By contrary in NYSE, the correlation relation mainly reflects the capital chain between financial enterprise, and the classification by community detection corresponds quite well to the standard industry classification.This paper's research methodology and conclusion will help people to deepen the understanding of differences between the Chinese and America stock market, providing an effective reference to the market macroscopic problem and investment portfolio.