摘要
作为现代金融理论的基石，有效市场假说对金融理论的发展起着至关重要的作用。有效市场假说把市场当作一个线性孤立的系统，投资者对市场信息的反应是线性的，然而大量实证研究显示：市场并非一直都处于均衡状态，有时市场也会发生动荡甚至崩溃。不同于有效市场假说，分形市场假说则认为市场是一个非线性、开放、耗散的系统，投资者对市场信息的反应是非线性的。因此，可以说有效市场假说只是分形市场假说的一个特例。在分形市场假说中，市场被认为是同时具备整体的确定性与局部的随机性，市场的分形结构可以揭示出价格波动的动力学特征。
分形理论与小波理论在尺度性能上具有很多相似性，所以小波理论非常适合刻画系统的分形特性。本文从分形理论、多重分形理论以及小波理论出发，详细阐述了基于小波理论的分形分析方法，并首次提出二分递归小波变换模极大值法（WTMM）来计算多重分形。随后文章应用这些理论，依次分析了股票市场的单重分形特性、多重分形特性，且以分析多重分形性的演化特征为主。在多重分形分析中，不仅采用了基于统计物理的配分函数法（PF）与基于数值分析的多重分形消除趋势波动分析法（MF-DFA），还采用了目前国际上广泛使用的小波分析法，包括小波变换模极大值法（WTMM）与小波领袖法（WL）。
首先，研究对中国股市的正态性进行了检验，并应用不同的方法对沪深股市的长期记忆性进行了考察，研究结果显示：中国股市的收益率序列具有较明显的“尖峰肥尾”特征，而所有方法计算得到的沪深股指的Hurst指数都大于0.5，说明沪深股指存在着正持续性；随着收益尺度增大，Hurst指数也逐渐增大，说明两市股指的长期收益具有更强的正持续性。
接着研究更多地考察了股票市场的多重分形特性，分为以下四个部分：
一、同时应用PF、MF-DFA考察了二十一世纪以来，中国股市与美、英、法、德、日五个主要股市的多重分形性，两种方法均显示：中国股市均显示出具有更强的多重分形性，与其它各股市相比，中国股指在低价位徘徊的时间更频繁，且大波动也较小波动更频繁。
二、基于多重分形消除趋势波动分析法MF-DFA，对日本七个经济时期以及中国股市自建立以来三个经济阶段的股票市场指数进行实证研究。研究结果显示：不同经济发展时期日、中两国的股票市场均具有明显的多重分形特性；但各自不同的经济时期多重分形特性差异显著，且与当时经济发展的状况存在着一定联系。最后，通过对照日中两国不同时期股票市场的多重分形性，得出一些对中国经济发展有益的启示。
三、与其他方法不同，小波变换模极大值法（WTMM）不但可以从数据自身结构侦测出系统的突变点，还可以基于突变点计算系统的多重分形特性。研究应用本文提出的二分递归小波变换模极大值法（WTMM）先通过建立道琼斯工业指数（DJI）与东京证交所股价指数（TPX）的模极大值线来定位金融危机发生的时点，然后选取道琼斯工业指数模极大值线上的奇异点系数对其进行多重分形分析。研究结果显示：小波变换模极大值法不仅可以准确定位金融危机发生的时点，还能刻画危机前后股市多重分形特性的变化。
四、研究通过应用小波领袖（WL）多重分析法刻画市场波动的多重分形特性来衡量市场的有效性，提出一种应用市场最大波动点集分形维数的演化来侦测金融风险发生时点的新方法，并与最大波动点集的奇异性指数结合起来对金融风险进行测量。研究结果表明：中、美、日三国在不同时期市场的有效性具有明显的差异，近年来中国市场有效性得到了显著提高，而美、日两国市场有效性则与金融风险的发生密切相关；此外，借助多重分形参数的演变能准确定位出金融风险发生的时点并对其大小进行测量。
综上所述，与基于均衡模型的有效市场假说理论相比，分形市场假说认为市场看作是一个复杂的非线性系统，所以它不仅可以刻画平稳运行时的市场，也可以考察市场在稳定与动荡之间的变换。对于市场的监督者与投资者来说，分形市场假说不仅有益于市场监管与投资决策，同时也有助于更有效地维护市场稳定与管理金融风险。
Being the foundation of modern finance theory, efficient market hypothesis is crucial to thedevelopment of finance theory. The market is considered to be a linear and isolated system inefficient market hypothesis and the information response of investors is linear, whereas lots ofempirical research shows that the market is not always under the equilibrium state and sometimesturbulence or even collapse would happen in the market. Different from the efficient markethypothesis, fractal market hypothesis considers the market as a nonlinear, open and dissipativesystem, and the information response of investors is nonlinear. Therefore, efficient markethypothesis is only a special case of fractal market hypothesis. In fractal market hypothesis，it isconsidered that the market has integraldeterminacyand localrandomicitysimultaneously, and thefractalstructureofmarket canrevealthedynamicalcharacteristicsofpricefluctuations.
Fractal theory and wavelet theory have many similarities of scale properties, so wavelettheory is very suit for describing the fractalcharacteristics ofsystem.This paper starts with fractaltheory, multifractal theory and wavelet theory, and then it elaborates the methods of analyzingfractal based on wavelet theory. The recursive dichotomy wavelet transform modulus maxima(WTMM) is first put forward in this paper. After that the paper applies the theory to analyze themonofractal and multifractal properties of stock markets, while the analysis of evolutionmultifractal characteristics is prior. In the multifractal analysis, not only partition function (PF)based onthestatisticalphysicsand multifractaldetrended fluctuationanalysis(MF-DFA) based onnumerical analysis are employed, but also the widelyapplied methods based on wavelet theory inthe world now are employed, including wavelet leaders (WL) and wavelet transform modulusmaxima(WTMM).
At first, normality test is implemented to China stock market and different methods areemployed to investigate the long-term memory of Shanghai and Shenzhen stock markets. Theresults shows that, the return rate series of China stock market have an apparent higher peak andfat-tailfeature,andHurstindexcalculatedbyallthemethodsaregreaterthan0.5,whichimplicatesthat positive persistence exists in Shanghai and Shenzhen stock market indices. Hurst index willincrease gradually as the scale ofreturn rate increases, which implicates that long-termreturn rateseriesdisplaystrongerpositivepersistence.
Andthen,thispaperpaysmoreattentiontothemultifractalpropertiesofstockmarkets,whichcanbearranged into fourparts.
Firstly, partition function(PF) and multifractal detrended fluctuation analysis (MF-DFA) areapplied to analyze the multifractal properties of China, US, England, France, Germany, Japanstock markets since the21
stcentury. The results show that China stock market has strongermultifractalproperties. Compared with theother stockmarkets, China stock market indexis morefrequent at low price levels, and the large fluctuations of index are more frequent than the smallfluctuations.
Secondly, based on the multifractal detrended fluctuation analysis (MF-DFA), the empiricalresearch are brought forward to Japanese stock market indices of the seven economy periods andChinese stock market indices of the three economy periods respectively. The result shows thatJapanese and Chinese stock market indices of all the economy periods have obvious multifractalproperties, which differ from each other significantly and have some relations with the differenteconomy status. At last, referring to the time-varying multifractal properties of Japanese andChinese stock markets, some beneficial implications for Chinese economy development areobtained.
Thirdly,different fromtheothermethods,therecursivedichotomywavelettransformmodulusmaxima(WTMM) methodcandetecttheoutliersofsystemand calculateits multifractalproperties.This method is employed to build up the maxima lines of DJI and TPX indices for detecting thetemporal locus of financial crisis firstly, and then it analyzes the multifractal properties of DJIbased onthecoefficientsofoutliers’onsome maxima lines. Theresultsshowthatthe method cannot only accurately locate the temporal locus of financial crisis, but also can characterize thevariationofstockmarket multifractalpropertiesbeforeandafterfinancialcrisis.
Fourthly, wavelet leaders (WL) multifractal analysis is employed to measure the marketefficiency by describing the multifractal properties of market fluctuations, then a new method isput forward to detect the temporal locus of financial crisis by fractal dimension evolution of thelargest fluctuations and measure the financial risk combined with singularity of the largestfluctuations.TheempiricalresultsshowthattheefficiencyofChina, UnitedStatesand Japanstockmarketsdiffer fromeachother apparentlyduring thedifferent periods, whichChinastockmarket’sefficiency has been improved significantly in the recent years while United States and Japan stock markets’efficiencyisrelatedwiththehappinessoffinancialcrisis. What’smore,thefinancialcrisiscanbedetectedand measuredpreciselybytheevolutionofthemultifractalparameters.
As a whole, compared with the efficient market hypothesis based on the equilibrium model,fractal market hypothesis considers the market as a complex and nonlinear system, so it can notonlydescribethestable market, but also can invest thetransitionofmarket betweenthestableandturbulent state. For the supervisors and investors, fractal market hypothesis is benefit for marketsupervision and investment decisions, and it can contribute to maintain the market stable andmanagethefinancialrisk moreefficiently.
引文
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