مدلسازی پویایی های قیمت سهام توسط سیستم نفوذ پیوسته و تجزیه و تحلیل سیستم های پیچیده مرتبط

The continuum percolation system is developed to model a random stock price process in this work. Recent empirical research has demonstrated various statistical features of stock price changes, the financial model aiming at understanding price fluctuations needs to define a mechanism for the formation of the price, in an attempt to reproduce and explain this set of empirical facts. The continuum percolation model is usually referred to as a random coverage process or a Boolean model, the local interaction or influence among traders is constructed by the continuum percolation, and a cluster of continuum percolation is applied to define the cluster of traders sharing the same opinion about the market. We investigate and analyze the statistical behaviors of normalized returns of the price model by some analysis methods, including power-law tail distribution analysis, chaotic behavior analysis and Zipf analysis. Moreover, we consider the daily returns of Shanghai Stock Exchange Composite Index from January 1997 to July 2011, and the comparisons of return behaviors between the actual data and the simulation data are exhibited.

The statistical analysis of financial market index and return is an active topic to understand and model the distribution of financial price fluctuation, which has long been a focus of economic research. As the stock markets are becoming deregulated worldwide, the modeling of dynamics of forwards prices is becoming a key problem in the risk management, physical assets valuation, and derivatives pricing, and it is also important to understand the statistical properties of fluctuations of stock prices in globalized securities markets. With the emerging of new statistical analyzing methods and computer-intensive analyzing methods in the last decade, the empirical results of stock returns provide various empirical evidence that has challenged the old random-walk hypothesis, requiring the invention of new financial models to describe price movements in the market. A series of statistical behaviors, the so-called “stylized facts”, such as fat tails phenomenon of price changes, power-law distributions of logarithmic returns and volume, volatility clustering of absolute returns and multifractality of volatility, are revealed from empirical research by previous studies [1], [2], [3], [4] and [5]. In an attempt to reproduce and explain these stylized facts, various market models have been introduced, some of which approach in this field by considering interacting particle systems [6], [7], [8], [9], [10], [11], [12] and [13]. Stauffer and Penna [8] and Tanaka [9] developed a price model by the lattice percolation system[14], [15], [16] and [17], the local interaction or influence among traders in a stock market is constructed, and a cluster of percolation is applied to define the cluster of traders sharing the same opinion about the market. They suppose that the spread of information leads to the stock price fluctuation, and when the influence rate of the model is around or at a critical value, the existence of fat-tail behavior for the returns is clearly observed. The critical phenomena of percolation model is used to illustrate the herd behavior of stock market participants. Zhang and Wang [12] invented the finite-range contact particle system to model a stock price process for studying the behaviors of returns by statistical analysis and computer simulation. The epidemic spreading of the contact model is considered as the spreading of the investors’ investment attitudes towards the stock market, and supposes that the investment attitudes are represented by the viruses of the contact model, which accordingly classify buying stock, selling stock and holding stock. In the present paper, we present a financial price model by applying the continuum percolation system. The continuum percolation model is usually referred to as a random coverage process or a Boolean model, it is a member of a class of stochastic processes known as interacting particle systems. In this financial model, the local interaction or influence among traders is developed by the continuum percolation, and a cluster of continuum percolation is applied to define the cluster of traders sharing the same opinion about the market. Then we study the power-law tails behavior, chaotic behavior and Zipf distribution of returns of the actual data and the simulation data by statistical analysis and computer simulation. We select the actual data for the closing prices of each trading day of Shanghai Stock Exchange Composite Index (SSE) in a fourteen and half-year period from 1 January 1997 to 29 July 2011, the total number of observed data for SSE is about 3525, and the database of SSE is from the website www.sse.com.cn.

In the present paper, we develop a stock price model by applying the theory of continuum percolation system to investigate the statistical behaviors of fluctuations of the stock prices. We discuss some statistical properties and power-law distributions of returns for different values of the percolation intensity λ, the length of market area l and the sight of the investor d. And we compare the return properties of the financial model with SSE composite index, including the fractal dimension, the largest Lyapunov exponent and the Zipf analysis. Through the comparison between the real market data and the simulated one, we hope to show that the financial model we present in this paper is reasonable to some extent and may be a new approach to study the statistical behaviors of stock market. The proposed financial model of this paper uses the formation of clusters in the continuum percolation theory to describe the herding behavior of investors in financial markets. One of the ways to aggregate interacting traders into clusters, or to imitate herding phenomena, is to assume that the connectivity between traders forming the groups can be seen as a pure geometrical percolation problem with fixed occupancy on a given network topology, for example see Refs. [1,36,37,7–9]. The percolation model is one of statistical physics systems, and the research of this paper belongs to a rapidly growing field, that of statistical finance, also called ‘‘econophysics’’. In this field, the concepts and methods of statistical physics are applied to study economic problems, and much recent work is focused on understanding the statistical properties of financial time series [38–40,6]. Stauffer and Sornette [7] introduced a self organized percolation model for stock market fluctuations. Rather than fixing the percolation connectivity artificially at or close to the critical value, as Cont and Bouchaud did, this model allows clusters to shatter and aggregate continuously as the connectivity parameter p evolves randomly. Makowiec et al. [37] extended the Cont–Bouchaud financial percolation model. The herd behavior is amplified by allowing clusters to copy decisions of some other cluster in the next time step. Jiang et al. [36] developed a financial price model for the cluster formation and information dispersal based on a scale-free network. In this model, the system is open and new investors continue to enter at each time step, resulting in the evolution of scale-free network, causing changes to the formation of clusters. Compared with previous work, our financial model has a major differences from these previous models in the formation of clusters of investors. In these previous researches, the investors are usually arranged on lattice vertices and then the clusters of investors are created by the random connectivity, whereas the proposed model of the present paper allows investors to randomly occupy the sites on the whole area, and the formation of clusters can be determined solely from the locations of the investors. For simplicity, our empirical research is made for the fixed parameters pu and pd, but there is a possibility to consider the changing parameters in the future study. We hope that this work can provide a different perspective in the formation of investor clusters under the price dynamics.