2022年2022年金融时间序列之分析[归 .pdf

AAPPS Bulletin April 2007, Vol. 17, No. 2What can We Learn from Analysis of the Financial Time Series?Bing-Hong Wang*1. INVESTIGATION OF THE DIS-TRIBUTION AND SCALING OF FLUCTUATIONS FOR STOCK INDEX IN FINANCIAL MAR-KETIn order to probe the extent of universality in the dynamics of complex behavior in financial markets and to provide a basic and appropriate framework for developing economic models of financial markets, we investigated the distribution of the fluctua -tions in the Hang Seng index the most important financial index in the Hong Kong stock market 1. The data include minute by minute records of the Hang Seng index from January 3, 1994 to May 28, 1997. It was observed that the distribution of returns in the Hang Seng index shows apparent scaling behavior, which cannot be modeled by a normal distribution. The non-Gaussian dynamics of the stochastic process underlying the time series of returns of the Hang Seng index, is better modeled by a truncated L vy distribution which is shown in Fig. 1. A power-law behavior is observed for the probability of zero return for time intervals ?t spanning at least two orders of magnitude. However, the power-law fall-off behavior in the tails deviate from that of L vy stable process. The two tails of the distribution drop more slowly than a Gaussian, but faster than a L vy process with an exponent outside the L vy stable region. Especially after remov-ing daily trading pattern from the data, the exponential deviation behavior from L vy stable process is more clearly. The daily pattern thus affects strongly the analysis of the asymptotic behavior and scaling of fluctuation distributions. The exponential truncation ensures the existence of a finite second moment. The observations are use-ful for establishing dynamical models of the Hong Kong stock market 1. 2. BUILD A FINANCIAL MARKET MODEL BASED ON SELF-OR-GANIZED PERCOLATIONThe economy has been perceived as a col-lection of nonlinear interacting units. This collection is complex; everything depends on everything else. Physicists are looking for empirical laws that can reveal such complex interactions and theories that will help understand them 2-5. As far as the financial markets are considered, due to intensive statistical studies during the last decade, the model of market fluctuation proposed by Bachelier in 1900 suffers the impugnation and the challenge of actual financial data such as the real-life markets are of return distributions displaying peak-center and fat-tail properties 6-7, one can observe volatility clustering and a non-triv-ial “ multifractal” scaling 8-10, and so on. These universal features portray a world of non Gaussian random walks and inspire scientists to construct microstructure market models, such as Cont-Bouchaud model 11, Lux-Marchesi model 12, LeBaron model 13 and so on, to explain its underlying mechanisms. Furthermore, this key problem about what is underlying market mechanisms, is still open.We focus on it for years and reap large profits from establishing and analyzing our market models including a activating What Can We Learn From Analysis of Financial Time SeriesDepartment of Modern Physics University of Science and Tech-nology of ChinaHefei, Anhui, 230026 China and Shanghai Academy of System Science Shanghai, 200093 ChinaE-mail: Bing-Hong WangHere we report the research work about analysis of the financial time series based on nonlinear dynamics and statistical physics undertook in recent years by USTC complex system research group.*名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 1 页，共 6 页 - - - - - - - - - AAPPS Bulletin April 2007, Vol. 17, No. 2Analysis and Modeling of Complex Time Seriesmodel of individual behavior towards eco-nomics complex system and a stock market based on “ Genetic Cellular Automata” with information exchange among individuals 14-15. Based on them, considering the self-organized dynamical evolution of the behavior of investors and their structure, we build an agent based model to describe financial markets. It has incorporated the following components: (1) the behavior of investors evolve constantly according to excess demand; (2) As reality, the circle of professionals and colleagues to whom a trader is typically connected evolves as a function of time: in some cases, traders follow strong herding behavior and their effective connectivity parameter p is high; in other cases, investors are more individu-alistic and smaller p seems more reason-able. So investors structure (the complex interactions between traders) undergoes generational metabolism process repeat-edly; (3) The effect of “ herd behavior” on the trade-volume and the impact of each invest-cluster s trade-volume on the price are nonlinear. While this artificial stock market evolving, the number of investors participating in trading isn t constant; the network made up of invest-clusters takes on different structure; cooperation and conflic-tion among invest-clusters are always op-erating; the affection of the herd behavior on the trade-volume varies dynamically accompanying the evolutionary of investor structure. In a word, the financial market is perceived as a complex system in which the large-scale dynamical properties depend on the evolutionary of a large number of nonlinear-coupled subsystems. This model can iterate for a period of any length. More simulations have been done indicating that the return distribution of the present model obeys L vy form in the center and displays fat-tail property, in accord with the stylized facts observed in real-life financial time series. Further -more, this model reveals the power-law relationship between the peak value of the probability distribution and the time scales in agreement with the empirical studies on the Hang Seng Index 16. It also achieves same avalanche dynamics and multi-fractal Fig. 1: Probability distributions of the returns and their scaling behavior of the Hang Seng index in Hong Kong stock market for the period January 3, 1994 to May 28, 1997. (a) The probability distributions of index returns for time separation ?t = 1, 2, 4, 8, 16, 32, 64, 128 min. (b) The central peak value P (0) as a function of ?t. A power-law behavior is observed. The slope of the best-fit straight line is 0.618 0.025 from which we obtain the scaling exponent = 1.619 0.05 characterizing the L vy distribution. (c) Re-scaled plot of the probability distributions shown in (a). Data collapse is evident after using rescaled variables with = 1.619. The abscissa is for the re-scaled returns, the ordinate is the logarithm of re-scaled probability.-0.200.20.40.60.811.21.41.61.822.22.42.62.8lg t-2.4-2.2-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4lgP(0):-40-30-20-10010203040Zs-4.5-4-3.5-3-2.5-2-1.5-1-0.5lgPs(zs) t=1 min t=2 min t=4 min t=8 min t=16 min t=32 min t=64 min t=128 min-300-200-1000100200300z-4-3.5-3-2.5-2-1.5-1-0.5lgProbability(z)t=1 mint=2 mint=4 mint=8 mint=16 mint=32 mint=64 mint=128 minWhat Can We Learn From Analysis of Financial Time Series名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 2 页，共 6 页 - - - - - - - - - AAPPS Bulletin April 2007, Vol. 17, No. 25scaling properties of price changes as the real 17-18. All the results indicate that un -derlying market mechanisms maybe is the self-organized dynamical evolution of the behavior of investors and their structure.3. MODELING STOCK MARKET BASED ON GENETIC CELLU-LAR AUTOMATAIn the paper 14, an artificial stock mar -ket based on genetic cellular automata is established. Cells are used to represent stockholders, who has the capability of self-teaching and are affected by the invest-ing history of the neighboring ones. The topological structure of CA in this paper is a two-dimensional square lattice with periodic boundary conditions. Before a trade, each stockholder should choose the trading strategies: to buy, to sell or to ride the fence. The stockholder s decision includes two steps: first, each stockholder works out a preparatory decision according to the his-tory of its investment and the stock price. The stockholders of different risk-proper-ties have different decision methods. The risk-neutral individuals directly inherit the last decision. The risk-aversed individuals investing strategy is to buy at a low price and to sell at a high price. The individuals risk-properties are given randomly initially, and can change along with the evolvement of the stock market. Similar to genetic algorithm 19, the risk property and the decision of each individual are naturally divided into four types of genes logically, including risk-property, deal-decision, price-decision and amount-decision. Each individual prefer to choose one of its successful neighbors to do crossover operation. The buyer with higher price and the seller with lower price will trade preferentially, and the trading-price is the average of seller s and buyer s price. The stock price is the weighted average of trad-ing-price according to the trading-amount. Simulation results about time series of price and returns based on genetic cellular automata show that when the proper initial condition and parameters have been cho-sen, the artificial stock market can generate its stock price whose trend and fluctuations are rather similar to that of real stock mar-If the former strategies are success, the clusters would stay the same strategies with a higher probability. Otherwise, they would change their strategies with a higher probability. Not only the former strategies would affect the clusters status, but also the neighbor clusters status would have an effect. The bigger the cluster is, the higher influence on neighbor it owns. If the neighbor clusters take the same strategies, they would form a bigger cluster. On the other hand, the bigger cluster would have a higher probability to collapse. Because in the real world, the market would become much more danger when everyone takes the same strategy. Finally, the price would be determined by the overall status of the clusters. If the number of the clusters which take “ buying” is bigger than “ selling” , the price increases, vice versa. In the next itera-tion, the fluctuation of the price would have a feedback of the clusters new status.The simulation results which agree well with the reality are convincing support to our original ideas to some extent 20. From the analysis based on this model, we learn that the causes of the various statistical properties of the real market are: the dy-namical evolvement of the trader groups, i.e., the process in which different trader groups cooperate and conflict; the gradu -ally accumulating process of the clusters growth controlled by the model automati-cally; the self-organized accumulating ef-fect on the magnified process of the “ herd behavior” . 5. EMPIRICAL STUDY ON THE VOLATILITY OF THE HANG-SENG INDEXThe volatility quantifies the activity of stock markets, defined as the number of transaction per unit of time connecting with interest of trades and is also the key input of virtually all option pricing models, including the classic Black and Scholes model 21 and the Cox-Ross-Rubinstein binomial model 22 that are based on the estimates of the asset volatility over the remaining life of the option. Without an efficient volatility estimate, it would be difficult for trades to identify situation in ket 14. In addition, in accordance with the empirical study on S&P500 7 and Hang Seng index 1, the central part of the probability distribution of price returns in this model can be well fitted by a Levy distribution, while its tail is really fat as shown in Fig. 1 1, 14.4. EVOLUTIONARY PERCOLA-TION MODEL OF STOCK MAR-KET WITH VARIABLE AGENT NUMBERThe financial market has been proved to be a very important platform for the research of the “ Complex System” field. By detailed analysis of the financial market price, more and more universal properties which are similar to those observed in physical systems with a large number of interact-ing units are discovered. The motivation to capture the complex behavior of stock market prices and market agents leads lots of sophisticated models based on different theoretical principles and evolutionary mechanics such as behavior-mind model, dynamic-games model, multi-agent model and so on. All of these models could repro-duce some of the stylized observations of real markets, but fail to account for either the origin of the universal characteristics or some very important properties of the real multi-agent system.To solve these problems, our new model is based on the following principles which were ignored by the former research 20: (1) The system is an open system which allows the agents get into or get out of it; (2) The growth of the clusters is a gradu-ally accumulating process controlled auto-matically by the model itself. (3) The “ herd behavior” is magnified by self-organized accumulating rather than by adjusting the parameters forcibly; (4) By cooperation or conflict, the clusters could get even bigger or crushed.To initialize our model, a lattice is taken up randomly. The interconnected nodes form a cluster. Every cluster could have three strategies: buying, selling and sleep -ing. During every iteration, some new agents would get into this system first. What Can We Learn From Analysis of Financial Time Series名师资料总结 - - -精品资料欢迎下载 - - - - - - - - - - - - - - - - - - 名师精心整理 - - - - - - - 第 3 页，共 6 页 - - - - - - - - - 6AAPPS Bulletin April 2007, Vol. 17, No. 2Analysis and Modeling of Complex Time Serieswhich options appear to be underpriced or overpriced. We study the statistical proper-ties of volatility of minite-by-minite price fluctuation of Hang-Seng index in Hong Kong stock market 23.The volatility is measured by locally averaging over a time window, the absolute value of price change over a short time in-terval. Define the price change as the differ-ence between two successive logarithms of the index. In our work 23, we have found that the cumulative distribution of volatility is consistent with the asymptotic power-law behavior, characterized by power exponent =2.12, different from previous studies as =3. The volatility distribution remains the same asymptotic power-law behavior for the time scale from ?t =10 min to ?t = 80 min. We can find that Hong Kong stock market is more uncontrolled for the investors compared with other stock markets, thus the concussion (or impact) introduced into this market by the investors is stronger. Furthermore, we investigated the volatility correlations via the power spectrum and DFA (detrended fluctuation analysis) after filtering the effect caused by the daily oscillation pattern. Both of these two methods convincingly demonstrate the existence of long-range correlations. These scaling properties of the volatility distribu-tion suggest that the volatility correlation is a possible explanation of the scaling behavior for price change distributio