证券市场的分形特征研究
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摘要
随着非线性科学和复杂性科学的蓬勃发展,越来越多的学者运用分形理论、混沌理论对金融证券市场价格波动中的非线性现象进行研究。本文将分形理论和小波理论相结合,对证券市场的分形结构和多重分形结构进行了深入研究。本文包括如下四个部分。
第一部分:证券市场的单分形特征研究。在金融市场中由于有效市场假说的不足和缺陷,分形市场理论被提出。分形市场理论主要研究金融时间序列的长记忆性和Hurst指数。本文运用R/S分析法、修正R/S分析法和V/S分析法对世界上主要的28个国家或地区的股票指数进行了长记忆性检验,并计算其Hurst指数。
第二部分:小波理论在单分形研究中的运用。基于最大重复离散小波变换的小波方差具有依尺度分解随机过程方差的特点,据此可以计算出金融序列的长记忆性参数。本文对上证指数、深证指数、美国标准普尔500指数进行最大重复离散小波变换,计算其长记忆参数。研究结果表明:1、选用不同的小波计算股票指数序列的小波方差,其结果相差很小;2、中国股市比美国的波动性强。
第三部分:证券市场的多重分形特征研究。本文对多重分形谱的计算方法进行了深入研究,首次提出了多重分形消除趋势波动分析(MFDFA)的改进方法——自适应多重分形消除趋势波动分析。原有的MFDFA法在计算前必须先固定去趋势拟合多项式的次数,而自适应MFDFA方法可以在计算时动态地调整拟合多项式的次数,改善了计算结果。本文分析了多重分形谱的两种计算方法——配分函数法和多重分形消除趋势波动分析的计算可靠性。使用这两种方法对Cantor二分集和三分集进行多重分形谱计算,将其与理论结果比较发现,在计算所选参数与Cantor集结构一致时结果很好,否则可能有偏差。使用两种方法对上证指数和美国标准普尔500指数进行了实证研究。
第四部分:小波理论在多重分形研究中的应用。首先分析了计算多重分形谱的小波变换模极大方法,并使用小波变换模极大方法对Cantor二分集和三分集进行多重分形谱计算,发现计算值与理论值符合很好,且不受参数值设置的影响。其次使用小波变换模极大方法对上证指数、深证成指和三支股票青岛海信、青岛啤酒、青岛海尔五分钟高频数据进行了多重分形谱的分析,并与MFDFA法的计算结果进行比较。最后研究了上证指数和深证指数的多重分形成因。对原始收益率序列进行置乱操作和替代数据处理,发现收益率序列的分布特征是多重分形谱产生的重要原因。在这部分本文提出了股票数据多重分形特征值的概念,可以用一对多重分形特征值来表征股票序列的多重分形特性。
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.
第一部分:证券市场的单分形特征研究。在金融市场中由于有效市场假说的不足和缺陷,分形市场理论被提出。分形市场理论主要研究金融时间序列的长记忆性和Hurst指数。本文运用R/S分析法、修正R/S分析法和V/S分析法对世界上主要的28个国家或地区的股票指数进行了长记忆性检验,并计算其Hurst指数。
第二部分:小波理论在单分形研究中的运用。基于最大重复离散小波变换的小波方差具有依尺度分解随机过程方差的特点,据此可以计算出金融序列的长记忆性参数。本文对上证指数、深证指数、美国标准普尔500指数进行最大重复离散小波变换,计算其长记忆参数。研究结果表明:1、选用不同的小波计算股票指数序列的小波方差,其结果相差很小;2、中国股市比美国的波动性强。
第三部分:证券市场的多重分形特征研究。本文对多重分形谱的计算方法进行了深入研究,首次提出了多重分形消除趋势波动分析(MFDFA)的改进方法——自适应多重分形消除趋势波动分析。原有的MFDFA法在计算前必须先固定去趋势拟合多项式的次数,而自适应MFDFA方法可以在计算时动态地调整拟合多项式的次数,改善了计算结果。本文分析了多重分形谱的两种计算方法——配分函数法和多重分形消除趋势波动分析的计算可靠性。使用这两种方法对Cantor二分集和三分集进行多重分形谱计算,将其与理论结果比较发现,在计算所选参数与Cantor集结构一致时结果很好,否则可能有偏差。使用两种方法对上证指数和美国标准普尔500指数进行了实证研究。
第四部分:小波理论在多重分形研究中的应用。首先分析了计算多重分形谱的小波变换模极大方法,并使用小波变换模极大方法对Cantor二分集和三分集进行多重分形谱计算,发现计算值与理论值符合很好,且不受参数值设置的影响。其次使用小波变换模极大方法对上证指数、深证成指和三支股票青岛海信、青岛啤酒、青岛海尔五分钟高频数据进行了多重分形谱的分析,并与MFDFA法的计算结果进行比较。最后研究了上证指数和深证指数的多重分形成因。对原始收益率序列进行置乱操作和替代数据处理,发现收益率序列的分布特征是多重分形谱产生的重要原因。在这部分本文提出了股票数据多重分形特征值的概念,可以用一对多重分形特征值来表征股票序列的多重分形特性。
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.
引文
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