پژوهش در ساختار فراکتال در بازار بورس چینی

Applying fractal theory, this paper probes and discusses self-similarity and scale invariance of the Chinese stock market. It analyses three kinds of scale indexes, i.e., autocorrelation index, Hurst index and the scale index on the basis of detrended fluctuation analysis (DFA) algorithm and promotes DFA into a recursive algorithm. Using the three kinds of scale indexes, we conduct empirical research on the Chinese Shanghai and Shenzhen stock markets. The results indicate that the rate of returns of the two stock markets does not obey the normal distribution. A correlation exists between the stock price indexes over time scales. The stock price indexes exhibit fractal time series. It indicates that the policy guide hidden at the back influences the characteristic of the Chinese stock market.

Scale invariance is an important theory and method to investigate the finance complexity problem. According to scale invariance, we may investigate the complexity of the time series in prices of stock markets and probe the laws of market fluctuations, i.e., scaling in the Norwegian stock market [1], a dynamic model describing stock market price distributions [2], an empirical study on the levy distribution of the stock market risk trace. In the middle of the 1990s, Peng et al. proposed a detrended fluctuation analysis (DFA) on the basis of applying DNA to test the correlation of long nucleotide chains, that is DFA method [4], [5] and [6]. The method is used to calculate the scale index in the complexity problem. It can analyse dynamic economic targets and stock market fluctuations. Empirical examples research on scaling in the English, Italian, Hungarian and Japanese stock markets and verify the degree of scaling in every market [7], [8], [9] and [10]. In [11], [12] and [13], the scaling in the Chinese stock market is analyzed, which is the basis of probing the laws of stock market fluctuations. First, it gives the characteristic targets describing the financial data. Second, from the angle of self-similarity and scale invariance, applying autocorrelation index, Hurst index and the scale index on the basis of DFA algorithm, we have an empirical analysis on the Chinese stock market. The results indicate that the rate of returns of the Shanghai and Shenzhen stock markets does not obey the normal distribution and exhibit a fractal time series and does not satisfy the efficient market hypothesis. Finally, it analyzes the fractal structure in the stock market, while promoting DFA into a recursive algorithm, which can improve calculation efficiency.

The empirical calculations of the self-similarity and the scale index of the Chinese stock market are as shown above. The fundamental results are as follows: (1) The fundamental statistics analysis. From Fig. 1, we can know that the rate of return of the stock indexes of the Shanghai and the Shenzhen markets have non-negative skewness. Compared with the normal distribution, the kurtosis of the rate of return is much larger than 3, exhibiting leptokurtic. The vertical square Fig. 1 and Fig. 4 of the standard deviation show strange values, which are the marks of the policy regulation. The frequency curves of the standard deviation in Fig. 2 and Fig. 5 exhibit the skewness distributions, and typical high peaks and heavy tails appear. The kurtosis of the Shanghai market is larger than that of the Shenzhen market. Comparing Figs. 3 to 6, the deviation divergence in the Shanghai stock market is larger than that of the Shenzhen market. It denotes that the stock price fluctuations in the Shanghai stock market are larger than those of the Shenzhen stock market. (2) Hurst index analysis. H=0.6965, the fractal dimension α=1.4358 in the Shanghai stock market. H=0.6866, the fractal dimension α=1.4564 in the Shenzhen stock market. Both the fractal dimensions in the two markets are less than 2. The markets have the character of the fractal structure and exhibit the characteristic of a long-term memory system which is generated by the non-linear random walk process. (3) Scale index analysis. In Fig. 11, the scale index of the Shanghai market is c=0.6875. In Fig. 12, the scale index of the Shenzhen market is c=0.6720. The scale indexes of the two markets are both larger than 0.5. The stock price index is the fractal time series and has the fractal structure characteristic. From Fig. 11 and Fig. 12 of the scale index, we can conclude that the scale index begins to decrease when it exceeds a certain time scale t. It denotes that the time series of the stock price index has limited the memory for the original information. Beyond it, the memory for the original information will be lost. (4) Autocorrelation index analysis. When the time series of the stock price index is a random series, according to Eq. (12), the estimated standard deviation by the autocorrelation function of the Shangzheng index is (1/2928)1/2=0.0185. The estimated standard deviation by the autocorrelation function of the Shencheng index is (1/2876)1/2=0.0186. From Fig. 8 and Fig. 10, we can know that the standard deviation increases the increase in View the MathML source. The autocorrelation index of the Shanghai combined index is γ=0.8491. The autocorrelation index of the Shencheng index is γ=0.8571. The autocorrelation indexes in the two markets are both less than 1. It denotes that the stock price index has a series correlation. The dependence of the adjacent observations of the stock price index is an important characteristic of the Chinese stock market. (5) Computation efficiency analysis. To analyse the Shanghai stock market fluctuations, comparing the computation time of the Hurst index with that of the Hurst index on the basis of the standard deviation in a 586 computer, it is, respectively, 48 and View the MathML source. In the computation methods of the scale index, the computation time of DFA is View the MathML source and the computation time of DFA recursive algorithm is View the MathML source, which denotes the computation efficiency of the two algorithms. The scaling exists extensively in the natural and financial systems. It may recover the complexity of the fluctuations and internal law of the stock market. Through the discussion of high peaks and heavy tails of the rate of return distributions, people generally accept the point that there exist high peaks in the distributions. But the argument of whether the random walk theory is shaken badly is very hot and strong. The paper applies three kinds of scale indexes to the empirical research on the Chinese stock market, and points out that the market has the characteristic of a fractal structure. It is the reference of the research on the complexity of the Chinese stock market [3], [17] and [18].