中国股票市场交易持续期的分形分析
|
摘要
本文通过对上海证券交易所的20支股票在2010年1月4日～2010年4月30日期间的共79个交易日的交易持续期数据,采取包括单分形和多分形等在内的多种分形分析方法进行了分析,描述了交易持续期在多个方面的特征。
在对本文选取的数据的基本统计特征进行分析之后,我们发现：交易持续期表现出和其他金融高频数据同样的非正态性,“日内效应”以及非稳定性。这些基本的统计特征的描述提示我们,可以采用分形分析方法对交易持续期序列进行分析,可以更好更精确地描述该序列的长期记忆性以及某些局部特征。
在使用单分形分析方法中的趋势消解分析法对交易持续期序列进行分析后,我们发现：交易持续期序列中确实存在着明显的长期记忆性,并且这种长期记忆性的存在是一种普遍现象,和股票所在行业、股票所属公司总流通市值以及总平均交易强度等指标没有直接的相关关系。另外,我们对交易持续期序列分别进行了三种不同的处理方式：去除“日内效应”、重排和傅立叶相位随机化,通过对处理后的数据的分析结果进行比较,可以看出,“日内效应”的存在对于交易持续期序列的长期记忆性并没有直接影响,长期记忆性的存在来自于序列的自身相关性,因为经过打乱重排后的序列并没有表现出明显的长期记忆性。同时我们发现,交易持续期的波动函数图像显示出斜率显著不同的两段,这表明在不同的时间尺度上,交易持续期序列的长期记忆特征也是有所不同的,也就是说,我们如果使用单一的时间尺度上的指标去量化整个序列的长期记忆特征是不甚科学和精确的。所以我们在下文中,选取了三种不同的多重分形分析方法,对交易持续期序列的多重分形特征进行了描述。
多重分形分析方法的实证结果表明,交易持续期序列中确实存在着明显的多重分形特征,在不同时间尺度上,序列表现出显著不同的特征。为了对产生这种多重分形特征的原因进行分析,我们对原始数据采用了不同的处理方法,并对处理后的数据进行分析和比较,我们发现,交易持续期序列的多重分形特征主要来自于序列的内在相关性以及序列的非正态性,并且后者对多重分形特征的产生具有更大的影响。
通过对交易持续期序列的单分形和多重分形分析后,我们更好地认识了中国股票市场中交易活动之间的一些内在关系,这对于我们理解和解释实际的股票交易活动具有很好的现实意义。
In this thesis, we use kinds of fractal analysis methods, including monofractal and multi-fractal, to describe the intertrade durations'characteristics in many areas, based on the data of the 20 stocks in Shanghai Stock Exchange, in 79 trading days from January 4,2010 to April 30,2010. After analyzed the basic statistical characteristics of the intertrade durations, we found that, the intertrade durations performance long-range memory, intraday pattern and non-stability, as the other financial ultra-high frequency data. These basic descriptions of the statistical characteristics inform us that fractal analysis methods could be used to describe the long-range memory and some other local features of the intertrade duration much better and more accurately.
In the using of the Detrended Fluctuation Analysis method, one of the monofractal analysis methods, to analysis of the data of intertrade duration we found that, there exists clear long-range memory in the series of intertrade duration, and the existence of such long-range memory is a common phenomenon, it dependent with these factors, such as the industry sector which the stocks in, total market capitalization and the total average trading intensity. In addition, the original data of the intertrade duration were carried out in three different approaches:removing the intraday pattern, shuffling and Fourier-phase randomization, the results of these different process upon the original data were compared, we can see that, the existence of the intraday pattern barely affects the long-range memory of intertrade duration, the long-range memory comes from the internal relevant of the series, because the series after shuffled doesn't performances any significant long-rang memory. At the same times, we discover that, the curves of the fluctuation functions of the intertrade durations constitutes with two lines with different slopes, these suggest that in different time scales the long-range memory of the intertrade durations appear differently, so we conclude that it's not reasonable and accuracy to quantify the long-range memory in different time scales of the whole series. Therefore, we use three different multi-fractal analyses method to describe the multi-fractal feathers of the intertrade durations.
Multi-fractal analysis of the empirical results show that in the data of intertrade duration there exists clear multi-fractal feathers, in different time scales, the series show significantly different characteristics. In order to find the cause of the appearance of these multi-fractal feathers, we processed the original data using different approaches, and compare the analysis result, we found that the multi-fractal feathers of the intertrade durations mainly come from the inherent correlation within sequence and the non-normality of the sequence, and the latter has a greater impact on the existence of the multi-fractal characteristics.
Through the monofractal and multi-fractal analysis of the intertrade duration, we have a better understanding of the intrinsic relationship between different trades in China's stock market. It's very useful for us to understand and explain the real activity of exchange in stock market.
在对本文选取的数据的基本统计特征进行分析之后,我们发现：交易持续期表现出和其他金融高频数据同样的非正态性,“日内效应”以及非稳定性。这些基本的统计特征的描述提示我们,可以采用分形分析方法对交易持续期序列进行分析,可以更好更精确地描述该序列的长期记忆性以及某些局部特征。
在使用单分形分析方法中的趋势消解分析法对交易持续期序列进行分析后,我们发现：交易持续期序列中确实存在着明显的长期记忆性,并且这种长期记忆性的存在是一种普遍现象,和股票所在行业、股票所属公司总流通市值以及总平均交易强度等指标没有直接的相关关系。另外,我们对交易持续期序列分别进行了三种不同的处理方式：去除“日内效应”、重排和傅立叶相位随机化,通过对处理后的数据的分析结果进行比较,可以看出,“日内效应”的存在对于交易持续期序列的长期记忆性并没有直接影响,长期记忆性的存在来自于序列的自身相关性,因为经过打乱重排后的序列并没有表现出明显的长期记忆性。同时我们发现,交易持续期的波动函数图像显示出斜率显著不同的两段,这表明在不同的时间尺度上,交易持续期序列的长期记忆特征也是有所不同的,也就是说,我们如果使用单一的时间尺度上的指标去量化整个序列的长期记忆特征是不甚科学和精确的。所以我们在下文中,选取了三种不同的多重分形分析方法,对交易持续期序列的多重分形特征进行了描述。
多重分形分析方法的实证结果表明,交易持续期序列中确实存在着明显的多重分形特征,在不同时间尺度上,序列表现出显著不同的特征。为了对产生这种多重分形特征的原因进行分析,我们对原始数据采用了不同的处理方法,并对处理后的数据进行分析和比较,我们发现,交易持续期序列的多重分形特征主要来自于序列的内在相关性以及序列的非正态性,并且后者对多重分形特征的产生具有更大的影响。
通过对交易持续期序列的单分形和多重分形分析后,我们更好地认识了中国股票市场中交易活动之间的一些内在关系,这对于我们理解和解释实际的股票交易活动具有很好的现实意义。
In this thesis, we use kinds of fractal analysis methods, including monofractal and multi-fractal, to describe the intertrade durations'characteristics in many areas, based on the data of the 20 stocks in Shanghai Stock Exchange, in 79 trading days from January 4,2010 to April 30,2010. After analyzed the basic statistical characteristics of the intertrade durations, we found that, the intertrade durations performance long-range memory, intraday pattern and non-stability, as the other financial ultra-high frequency data. These basic descriptions of the statistical characteristics inform us that fractal analysis methods could be used to describe the long-range memory and some other local features of the intertrade duration much better and more accurately.
In the using of the Detrended Fluctuation Analysis method, one of the monofractal analysis methods, to analysis of the data of intertrade duration we found that, there exists clear long-range memory in the series of intertrade duration, and the existence of such long-range memory is a common phenomenon, it dependent with these factors, such as the industry sector which the stocks in, total market capitalization and the total average trading intensity. In addition, the original data of the intertrade duration were carried out in three different approaches:removing the intraday pattern, shuffling and Fourier-phase randomization, the results of these different process upon the original data were compared, we can see that, the existence of the intraday pattern barely affects the long-range memory of intertrade duration, the long-range memory comes from the internal relevant of the series, because the series after shuffled doesn't performances any significant long-rang memory. At the same times, we discover that, the curves of the fluctuation functions of the intertrade durations constitutes with two lines with different slopes, these suggest that in different time scales the long-range memory of the intertrade durations appear differently, so we conclude that it's not reasonable and accuracy to quantify the long-range memory in different time scales of the whole series. Therefore, we use three different multi-fractal analyses method to describe the multi-fractal feathers of the intertrade durations.
Multi-fractal analysis of the empirical results show that in the data of intertrade duration there exists clear multi-fractal feathers, in different time scales, the series show significantly different characteristics. In order to find the cause of the appearance of these multi-fractal feathers, we processed the original data using different approaches, and compare the analysis result, we found that the multi-fractal feathers of the intertrade durations mainly come from the inherent correlation within sequence and the non-normality of the sequence, and the latter has a greater impact on the existence of the multi-fractal characteristics.
Through the monofractal and multi-fractal analysis of the intertrade duration, we have a better understanding of the intrinsic relationship between different trades in China's stock market. It's very useful for us to understand and explain the real activity of exchange in stock market.
引文
[1]A. Yuen, P.C. Ivanov, Impact of Stock Market Structure on Intertrade Time and Price Dynamics [J], Physics (2005) 0508203.
[2]Admati A.R., Pfleiderer P. A Theory of Intraday Patterns:Volume and Price Variability [J]. The Review of Financial Studies,1988,1(1):3-40.
[3]Antonio Turiel, Conrad J. Perez-Vicente, Role of Multifractal Sources in the Analysis of Stock Market Time Series [J], Physica A 355(2005):475-496.
[4]Andersen T G., Bollerslev T, Answering the Critics:Yes, ARCH Models Do Provide Good Volatility Forecasts [J], International Economic Review, Vol.39, no.4 (1998), pp.885-905.
[5]Andersen T G., Bollerslev T., "Intraday Periodicity and Volatility Persistence in Financial Markets" [J], Journal of Empirical Finance, No.4 (1997), pp.115-158.
[6]Andersen T G., Bollerslev T, Cai J., " Intraday and Interday Volatility in the Japanese Stock Market" [J], Journal of International Markets, no.10 (2000), pp.107-130.
[7]Camilo Rodrigues Neto, Andre C. R. Martins, Multifractality in the Parameter Model for Multivariate [J], Physica A 388(2009):2198-2206.
[8]Diamond, D. W. and R. E. Verrecchia, Constraints on Short-selling and Asset Price Adjustments to Private Information[J], Journal of Financial Economics,1987,18:277-311.
[9]Easley D, O' Hara, Time and the Process of Security Price Adjustment [J]. Journal of Finance,1992,47 (2):905-927.
[10]Edgar E.Peters,分形市场分析——将混沌理论应用到投资与经济理论上[M].储海林等译.北京：经济科学出版社,2002.
[11]Edgar E.Peters,资本市场的混沌与秩序[M].王小东译.北京：经济科学出版社,1999.
[12]Guoxiong Du, Xuanxi Ning, Multifractal Properties of Chinese Stock Market in Shanghai [J], Physica A 387(2008):261-269.
[13]Gwilym O, Sutcliffe C., Problems Encountered When Using High Frequency Financial Market Data: Suggested Solutions [J]. Journal of Financial Management and Analysis,2001,14 (1):38-51.
[14]H. Demsetz, the Cost of Thansacting [J], Quarterly Journal of Economics,1968 82:33-53.
[15]Havlin S., Amaral Lan, Application of Statistical Physics to Heartbeat Diagnosis [J], Physica A,1999, 274:99-110.
[16]Hurst H.E., Long-term Storage of Reservoirs [J]. Trans. Amer. Soc. Civil Eng.,1951,116:770-808.
[17]Jose Alvarez-Ramirez, Jesus Alvarez, Eduardo Rodriguez, Guillermo Fernandez-Anaya, Time-varying Hurst Exponent for U.S. Stock Markets [J], Physica A 387(2008):6159-6169.
[18]Kantelhardt Jan W. Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series [J], Physica A,2002,316:87-114.
[19]Kettani, A Novel Approach to the Estimation of the Long Range Dependence Parameter, A Dissertation for the Digree of Ph. D,2002.
[20]L. Zunino, B. M. Tabak, A. Figliola, D. G. Perez, M. Garavaglia, O. A. Rosso, A Multifractal Approach for Stock Market Inefficiency [J], Physica A 387(2008):6558-6566.
[21]Mandelbrot B, Wallis J R. Robustness of the Rescaled Range R/S in the Measurement of Non-cyclic Long Run Statistical Dependence, Water Resour.1969, Res.5909.
[22]Mauro Politi, Enrico Scalas, Fitting the Empirical Distribution of Intertrade Durations [J], Physica A 387(2008):2025-2034.
[23]P. Oswiecimka, J. Kwapien, S. Drozdz, Multifractality in the Stock Market:Price Increments Versus Waiting Times [J], Physica A 347(2005):626-638.
[24]P.C. Ivanov, A. Yuen, B. Podobnik, Y. K. Lee, Common Scaling Patterns in Intertrade Times of U. S. Stocks [J], Phys. Rev. E 69 (2004) 056107.
[25]Peng C. K., Buldyrev S. V., Simons M., Stanley H. E., Goldberger A.L., Mosaic Organization of DNA Nucleotides [J]. Physical Review E49 (1994):1685-1689.
[26]Ramazan Gencay, Faruk Selcuk, Abdurrahman Ulugulyagci, High Volatility, Thick Tails and Extreme Value Theory in Value-at-Risk Estimation [J], Insurance:Mathematics and Economics 33(2003):337-356.
[27]Ruey S. Tsay,金融时间序列分析[M].潘家柱译.北京：机械工业出版社,2007.
[28]Ruipeng Liu, T. Di Matteo, Thomas Lux, True and Apparent Scaling:The Proximity of the Markov-switching Multifractal Model to Long-rang Dependence [J], Physica A 383(2007):35-42.
[29]Sergio Ledesma. Analysis and Synthesis of Self-Similar Network Traffic [J]. A Dissertation for the Digree of Ph. D,2000.
[30]Vadim Teverovsky, Murad S, Taqqu, Walter Willinger. A Critical Look at Lo's Modified R/S Statistic [J]. Journal of Statistical Planning and Inference,1999,80:211-227.
[31]Wei-Xing Zhou, Detrended Fluctuation Analysis of Intertrade Duration [J], Physica A 388(2009)433-440.
[32]Wei-Xing Zhou, Multifractal Analysis of Chinese Stock Volatilities based on the Partition Approach [J], Physica A 387(2008):4881-4888.
[33]Wei-Xing Zhou, Scaling in the Distribution of Intertrade Durations of Chinese Stocks [J], Physica A 387(2008):5818-5825.
[34]Wei-Xing Zhou, Scaling and Memory in the Return Intervals of Realized Volatility [J], Physica A 388(2009):4787-4796.
[35]Wilison P. S., Tomsett A. C, Toumi R., Long-memory Analysis of Time Series with missing Values[J].Physical Review E(2003),68(010703).
[36]Xia Sun, Huiping Chen, Yongzhuang Yuan, Ziqin Wu, Predictability of Multifractal Analysis of Hang Seng Stock Index in Hong Kong [J], Physica A 301(2001):473-482.
[37]Xia Sun, Huiping Chen, Ziqin Wu, Yongzhuang Yuan, Multifractal Analysis of Hang Seng Index in Hong Kong Stock Market [J], Physica A 291(2001):553-562.
[38]Yu Wei, Peng Wang, Forecasting Volatility of SSEC in Chinese Stock Market Using Multifractal Analysis [J], Physica A 387(2008):1585-1592.
[39]Yu Wei, Dengshi Huang, Multifractal Analysis of SSEC in Chinese Stock Market:A different Empirical Result from Hang Seng Index [J], Physica A 355(2005):497-508.
[40]Yuan Ying, Zhuang Xin-tian, Jin Xiu, Measuring Multifractality of Stock Price Fluctuation Using Multifractal Detrended Fluctuation Analysis [J], Physica A 388(2009):2189-2197.
[41]Yuan Ying, Zhuang Xin-tian, Multifractal Description of Stock Price Index Fluctuation Using a Quadratic Function Fitting [J], Physica A 387(2008):511-518.
[42]Z. Eisler, J. kertesz, Multifractal Model of Asset Returns with leverage Effect [J], Physica A 343(2004):603-622.
[43]Z. Eisler, J. Kertesz, Size Matters:Some Stylized Facts of the Stock Market Revisited [J], Eur. Phys. J. B 51 (2006) 145-154.
[44]曹宏铎,韩文秀,李昊.投资机会决策中分数布朗运动理论[J].系统工程学报.2001,1(16)：4549.
[45]陈保华,市场微观结构理论研究综述[J],经济学动态.2003.1.
[46]高铁梅,计量经济分析方法与建模——EViews应用及实例[M].北京：清华大学出版社,2005.
[47]耿克红等,中国股市超高频持续期序列长记忆性研究[J],中国管理科学,2008.4.
[48]耿克红,超高频数据下金融市场持续期模型研究[D],天津大学博士学位论文,2007：1-7.
[49]何建敏,常松.中国股票市场多重分形游走及其预测[J].中国管理科学.2002,10(3)：11-17.
[50]黄诒蓉,中国股市分形结构：理论与实证[M].广州：中山大学出版社,2006
[51]赖川波,中国股票市场时变的Hurst指数及多重分形分析[D],浙江工商大学硕士学位论文,2010.
[52]李平,曾勇,唐小我,市场微观结构理论综述明[J],管理科学学报.2003.5.
[53]李亚静等,中国股市收益率与波动性长记忆性的实证研究[J],系统工程理论与实践,2003.1.
[54]陶利赋等,中国股市高频数据中的周期性和长期记忆性[J],系统工程理论与实践,2004.6.
[55]魏莉娅,基于高频数据的基金市场长记忆性研究[D],电子科技大学硕士学位论文,2007.
[56]赵巍,分形分析方法在中国证券市场中的应用研究[D],东南大学硕士学位论文,2005.
[57]谢淑云,多重分形方法在金属成矿评价中的作用[J],成都理工大学学报,2004,1(31)：28-33.
[58]苑莹,庄新田,股票市场多重分形性的统计描述[J],金融管理(2007)：3-7.
[59]杨桂元等,中国股市收益率和波动率的长记忆性检验[J],统计与信息论坛,2009.6.
[60]张波等,基于高频数据的沪指波动长记忆性驱动因素分析[J],统计与信息论坛,2009.6.
[2]Admati A.R., Pfleiderer P. A Theory of Intraday Patterns:Volume and Price Variability [J]. The Review of Financial Studies,1988,1(1):3-40.
[3]Antonio Turiel, Conrad J. Perez-Vicente, Role of Multifractal Sources in the Analysis of Stock Market Time Series [J], Physica A 355(2005):475-496.
[4]Andersen T G., Bollerslev T, Answering the Critics:Yes, ARCH Models Do Provide Good Volatility Forecasts [J], International Economic Review, Vol.39, no.4 (1998), pp.885-905.
[5]Andersen T G., Bollerslev T., "Intraday Periodicity and Volatility Persistence in Financial Markets" [J], Journal of Empirical Finance, No.4 (1997), pp.115-158.
[6]Andersen T G., Bollerslev T, Cai J., " Intraday and Interday Volatility in the Japanese Stock Market" [J], Journal of International Markets, no.10 (2000), pp.107-130.
[7]Camilo Rodrigues Neto, Andre C. R. Martins, Multifractality in the Parameter Model for Multivariate [J], Physica A 388(2009):2198-2206.
[8]Diamond, D. W. and R. E. Verrecchia, Constraints on Short-selling and Asset Price Adjustments to Private Information[J], Journal of Financial Economics,1987,18:277-311.
[9]Easley D, O' Hara, Time and the Process of Security Price Adjustment [J]. Journal of Finance,1992,47 (2):905-927.
[10]Edgar E.Peters,分形市场分析——将混沌理论应用到投资与经济理论上[M].储海林等译.北京：经济科学出版社,2002.
[11]Edgar E.Peters,资本市场的混沌与秩序[M].王小东译.北京：经济科学出版社,1999.
[12]Guoxiong Du, Xuanxi Ning, Multifractal Properties of Chinese Stock Market in Shanghai [J], Physica A 387(2008):261-269.
[13]Gwilym O, Sutcliffe C., Problems Encountered When Using High Frequency Financial Market Data: Suggested Solutions [J]. Journal of Financial Management and Analysis,2001,14 (1):38-51.
[14]H. Demsetz, the Cost of Thansacting [J], Quarterly Journal of Economics,1968 82:33-53.
[15]Havlin S., Amaral Lan, Application of Statistical Physics to Heartbeat Diagnosis [J], Physica A,1999, 274:99-110.
[16]Hurst H.E., Long-term Storage of Reservoirs [J]. Trans. Amer. Soc. Civil Eng.,1951,116:770-808.
[17]Jose Alvarez-Ramirez, Jesus Alvarez, Eduardo Rodriguez, Guillermo Fernandez-Anaya, Time-varying Hurst Exponent for U.S. Stock Markets [J], Physica A 387(2008):6159-6169.
[18]Kantelhardt Jan W. Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series [J], Physica A,2002,316:87-114.
[19]Kettani, A Novel Approach to the Estimation of the Long Range Dependence Parameter, A Dissertation for the Digree of Ph. D,2002.
[20]L. Zunino, B. M. Tabak, A. Figliola, D. G. Perez, M. Garavaglia, O. A. Rosso, A Multifractal Approach for Stock Market Inefficiency [J], Physica A 387(2008):6558-6566.
[21]Mandelbrot B, Wallis J R. Robustness of the Rescaled Range R/S in the Measurement of Non-cyclic Long Run Statistical Dependence, Water Resour.1969, Res.5909.
[22]Mauro Politi, Enrico Scalas, Fitting the Empirical Distribution of Intertrade Durations [J], Physica A 387(2008):2025-2034.
[23]P. Oswiecimka, J. Kwapien, S. Drozdz, Multifractality in the Stock Market:Price Increments Versus Waiting Times [J], Physica A 347(2005):626-638.
[24]P.C. Ivanov, A. Yuen, B. Podobnik, Y. K. Lee, Common Scaling Patterns in Intertrade Times of U. S. Stocks [J], Phys. Rev. E 69 (2004) 056107.
[25]Peng C. K., Buldyrev S. V., Simons M., Stanley H. E., Goldberger A.L., Mosaic Organization of DNA Nucleotides [J]. Physical Review E49 (1994):1685-1689.
[26]Ramazan Gencay, Faruk Selcuk, Abdurrahman Ulugulyagci, High Volatility, Thick Tails and Extreme Value Theory in Value-at-Risk Estimation [J], Insurance:Mathematics and Economics 33(2003):337-356.
[27]Ruey S. Tsay,金融时间序列分析[M].潘家柱译.北京：机械工业出版社,2007.
[28]Ruipeng Liu, T. Di Matteo, Thomas Lux, True and Apparent Scaling:The Proximity of the Markov-switching Multifractal Model to Long-rang Dependence [J], Physica A 383(2007):35-42.
[29]Sergio Ledesma. Analysis and Synthesis of Self-Similar Network Traffic [J]. A Dissertation for the Digree of Ph. D,2000.
[30]Vadim Teverovsky, Murad S, Taqqu, Walter Willinger. A Critical Look at Lo's Modified R/S Statistic [J]. Journal of Statistical Planning and Inference,1999,80:211-227.
[31]Wei-Xing Zhou, Detrended Fluctuation Analysis of Intertrade Duration [J], Physica A 388(2009)433-440.
[32]Wei-Xing Zhou, Multifractal Analysis of Chinese Stock Volatilities based on the Partition Approach [J], Physica A 387(2008):4881-4888.
[33]Wei-Xing Zhou, Scaling in the Distribution of Intertrade Durations of Chinese Stocks [J], Physica A 387(2008):5818-5825.
[34]Wei-Xing Zhou, Scaling and Memory in the Return Intervals of Realized Volatility [J], Physica A 388(2009):4787-4796.
[35]Wilison P. S., Tomsett A. C, Toumi R., Long-memory Analysis of Time Series with missing Values[J].Physical Review E(2003),68(010703).
[36]Xia Sun, Huiping Chen, Yongzhuang Yuan, Ziqin Wu, Predictability of Multifractal Analysis of Hang Seng Stock Index in Hong Kong [J], Physica A 301(2001):473-482.
[37]Xia Sun, Huiping Chen, Ziqin Wu, Yongzhuang Yuan, Multifractal Analysis of Hang Seng Index in Hong Kong Stock Market [J], Physica A 291(2001):553-562.
[38]Yu Wei, Peng Wang, Forecasting Volatility of SSEC in Chinese Stock Market Using Multifractal Analysis [J], Physica A 387(2008):1585-1592.
[39]Yu Wei, Dengshi Huang, Multifractal Analysis of SSEC in Chinese Stock Market:A different Empirical Result from Hang Seng Index [J], Physica A 355(2005):497-508.
[40]Yuan Ying, Zhuang Xin-tian, Jin Xiu, Measuring Multifractality of Stock Price Fluctuation Using Multifractal Detrended Fluctuation Analysis [J], Physica A 388(2009):2189-2197.
[41]Yuan Ying, Zhuang Xin-tian, Multifractal Description of Stock Price Index Fluctuation Using a Quadratic Function Fitting [J], Physica A 387(2008):511-518.
[42]Z. Eisler, J. kertesz, Multifractal Model of Asset Returns with leverage Effect [J], Physica A 343(2004):603-622.
[43]Z. Eisler, J. Kertesz, Size Matters:Some Stylized Facts of the Stock Market Revisited [J], Eur. Phys. J. B 51 (2006) 145-154.
[44]曹宏铎,韩文秀,李昊.投资机会决策中分数布朗运动理论[J].系统工程学报.2001,1(16)：4549.
[45]陈保华,市场微观结构理论研究综述[J],经济学动态.2003.1.
[46]高铁梅,计量经济分析方法与建模——EViews应用及实例[M].北京：清华大学出版社,2005.
[47]耿克红等,中国股市超高频持续期序列长记忆性研究[J],中国管理科学,2008.4.
[48]耿克红,超高频数据下金融市场持续期模型研究[D],天津大学博士学位论文,2007：1-7.
[49]何建敏,常松.中国股票市场多重分形游走及其预测[J].中国管理科学.2002,10(3)：11-17.
[50]黄诒蓉,中国股市分形结构：理论与实证[M].广州：中山大学出版社,2006
[51]赖川波,中国股票市场时变的Hurst指数及多重分形分析[D],浙江工商大学硕士学位论文,2010.
[52]李平,曾勇,唐小我,市场微观结构理论综述明[J],管理科学学报.2003.5.
[53]李亚静等,中国股市收益率与波动性长记忆性的实证研究[J],系统工程理论与实践,2003.1.
[54]陶利赋等,中国股市高频数据中的周期性和长期记忆性[J],系统工程理论与实践,2004.6.
[55]魏莉娅,基于高频数据的基金市场长记忆性研究[D],电子科技大学硕士学位论文,2007.
[56]赵巍,分形分析方法在中国证券市场中的应用研究[D],东南大学硕士学位论文,2005.
[57]谢淑云,多重分形方法在金属成矿评价中的作用[J],成都理工大学学报,2004,1(31)：28-33.
[58]苑莹,庄新田,股票市场多重分形性的统计描述[J],金融管理(2007)：3-7.
[59]杨桂元等,中国股市收益率和波动率的长记忆性检验[J],统计与信息论坛,2009.6.
[60]张波等,基于高频数据的沪指波动长记忆性驱动因素分析[J],统计与信息论坛,2009.6.