Study On Fractal Characteristics In Securities Market

With the development of nonlinear science and complexity science,more and more scholars use fractal theory and chaos theory to research the nonlinear phenomena in price volatility of the financial market.The thesis combines the fractal theory and Wavelet theory to research the fractal structure and multifractal structure in securities market.This thesis contains the following four sections.Section 1 studies the monofractal of securities market.Due to the weakness of Efficient Market Hypothesis in financial market,the fractal market theory is proposed. The fractal market theory mainly studys long memory and the Hurst index of financial time series.The R/S method,the modified R/S method and V/S method are used to verify length memory of stock indexes of 28 countries or regions around the world and account theirs Hurst indexes.Section 2 applies the wavelet theory to monofractal.The wavelet variance based on Maximal Overlap Discrete Wavelet Transform can decompose stochastic process by scales,so long memory parameter is calculated according to this property.Shanghai stock index,Shenzhen stock index and American standard poor 500 index are transformed by Maximal Overlap Discrete Wavelet transform,and then their long memory parameters are accounted.Research shows that different wavelets have a little effect for Wavelet Variance of stock index,and Chinese stock market is more fluctuant than America stock market.Section 3 gives the research of the multifractal in securities market.This section studies the calculation method of multifractal spectrums,and proposes the adaptive muitifractal detrended fluctuation analysis(AMFDFA),which is an improvement of the previous multifractal detrended fluctuation analysis(MFDFA).The original MFDFA method must fix degree of fitting polynomial before calculating,but AMFDFA method can dynamically adjust degree of fitting polynomial during calculating and improve the result.This section analyzes the reliability of two methods of multifractal spectrums calculation that are partition function method and multifractal detrended fluctuation analysis,and the above methods are used to calculate multifractal spectrums of Cantor two-divided aggregate and three-divided aggregate.Comparing their results with theoretical value,it shows that if the chosen parameters are consistent with Cantor structure,the result is good,otherwise it may exist deviation.The above two methods are used to empirical research for Shanghai stock index and American standard poor 500 index.Section 4 applies the wavelet theory to multifractal.Firstly,this section analyzes Wavelet Transform Modulus Maxima(WTMM) that can calculate multifractal spectrums. WTMM method is used to calculate multifractal spectrums of Cantor two-divided aggregate and three-divided aggregate.Comparing their results with theoretical value,it shows that the result is good and parameters have no effect on the result.Next.WTMM is used to analyze multifractal spectrums of three stocks 5-minute high-frequency data of Qingdao Hisense.Qingdao beer and Qingdao Haier,and WTMM's results is compared with MFDFA's.Finally,the sources of multifractal spectrums of Shanghai stock and Shenzhen stock are analyzed.After original returns series are conducted by permuting and surrogate data,it is found that distributing character of returns is important source of multifractal spectrums.In this section,the concept of multifractal eigenvalue is proposed, so a pair of multifractal eigenvalue can represent multifractal character of stock series.