Synthesis Study Of Fractal&Wavelet And Applications In The Analysis Of Stock Market Fluctuations

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 the21stcentury. 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.