沪深股市分形结构实证分析
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摘要
股票市场价格受多种因素的影响具有不确定性,人们总是试图寻找一种理论来消除这种不确定性,达到预测股票价格的目的。从Fama提出有效市场假说(Efficient Market Theory,EMH)以来,便迅速成为建立和研究现代金融理论的基石,这种线性范式长期主宰金融经济学达半个世纪之久。
然而,随着股票市场的发展和人们对股票市场认识水平的提高,作为现代金融理论基石的有效市场假说(EMH)越来越多地被实践证明不符合现实,如一月效应、小公司效应等,因此许多学者试图寻找一种更好的理论来解释这些现象。原本用于研究海岸线的分形理论便被引入到股票市场的研究中,建立在非线性动力学基础上的分形市场理论,放松了有效市场假说的条件,很好的解释了有效市场假说无法解释的各种市场现象。以分形理论和方法来解释中国股票市场的非线性特征,有利于深入理解和认识股票市场的有效性、波动性和长期记忆性,具有非常重要的理论和现实意义。
本文选取上证综指和深圳成指为研究对象,分别选取成立以来至2010年7月1日收盘价数据和周收盘价数据,以Eviews和Matlab软件为工具,运用经典的R/S方法,对沪深股市收益率进行了实证分析,分别计算了沪深股市的H指数和周期长度。结果表明,沪深股市均具有明显的分形特征,其价格变化具有趋势性和循环性。本文的创新之处首先在于对以前的研究成果进行了系统的总结,包括有效市场理论和分形市场理论的概念、发展过程和应用等。然后选取了最新的有代表性的沪深股市周收益和日收益数据进行了实证分析,数据从沪深股市成立至今直至2010年7月1日,相比已有研究数据更多、更丰富。同时本文改变了以往时间序列分析采用Eviews软件的缺陷,采用功能更加强大的Matlab软件,得出的结论更加符合实际。
由于分形理论的可预测性,本文推导了分形布朗运动下的Black-Scholes模型,并应用于对股票价格的预测。最后本文分析了沪深股市非线性及分形特征的成因,并对分形理论的进一步研究做出了展望。
Affected by many factors, Stock market price has uncertainty. People are always trying to find a theory to eliminate this uncertainty, to achieve the purpose of forecasting stock prices. From Fama introduced the Efficient Market Theory (EMH), which quickly becomes the cornerstone of modern financial economic, this linear paradigm dominates financial economics for half a century.
However, with the stock market developping, and people raising the awareness of the stock market,as the cornerstone of mordern financial economic,the EMH is proved incompatible with the reality more and more,such as the first month phenomenon and the small company phenomenon. Therefore, many scholars tried to find a better theory to explain these phenomena.The Fractal Theory,which originally used to study the fractal coastline, has been introduced to the study of stock market. The Fractal Theory,which established on the basis of the nonlinear dynamics, relaxes the conditions of the efficient market hypothesis,gives a better explanation of the phenomenon that can not be explained by the efficient market hypothesis. Using the Fractal theory and method to explain the nonlinear characteristics of the Chinese stock market, conducive to understanding and awareness the effectiveness of the stock market, volatility and long-term memory, So it has very important theoretical and practical significance.
This artical selected the Shanghai and Shenzhen Composites as the object, selecting the daily closing price data and the weekly closing price data from the inception to July 1st,2010.using the Eviews and Matlab software and the classic R/S method,analysized the Shanghai and Shenzhen stock market returns,and calculated the H index and the cycle length of the Shanghai and Shenzhen stock market.The results show that:they are Fractal,and their price changes have trends and cyclical. The innovation of this paper is that,tfirst we summary the previous research results,which including the efficient market theory,the concept of fractal market theory and the application. Then we select the latest representative of the Shanghai and Shenzhen stock market returns and weekly earnings data do analysis,which from Shanghai and Shenzhen stock markets'establishment to July 1st,2010.Compared to the data have been,it's more richer. At the same time,we using the Matlab software to avoid the defacts of the Eviews,so the conclusion is more powerful and realistic.
Because of its predictability, this artical derived the Black-Scholes model under Fractional Brownian motion, and applied to the prediction of stock price. Finally, we also analysized the causes of the nonlinear and the fractal,and make a further research prospect of the Fractal Theory.
然而,随着股票市场的发展和人们对股票市场认识水平的提高,作为现代金融理论基石的有效市场假说(EMH)越来越多地被实践证明不符合现实,如一月效应、小公司效应等,因此许多学者试图寻找一种更好的理论来解释这些现象。原本用于研究海岸线的分形理论便被引入到股票市场的研究中,建立在非线性动力学基础上的分形市场理论,放松了有效市场假说的条件,很好的解释了有效市场假说无法解释的各种市场现象。以分形理论和方法来解释中国股票市场的非线性特征,有利于深入理解和认识股票市场的有效性、波动性和长期记忆性,具有非常重要的理论和现实意义。
本文选取上证综指和深圳成指为研究对象,分别选取成立以来至2010年7月1日收盘价数据和周收盘价数据,以Eviews和Matlab软件为工具,运用经典的R/S方法,对沪深股市收益率进行了实证分析,分别计算了沪深股市的H指数和周期长度。结果表明,沪深股市均具有明显的分形特征,其价格变化具有趋势性和循环性。本文的创新之处首先在于对以前的研究成果进行了系统的总结,包括有效市场理论和分形市场理论的概念、发展过程和应用等。然后选取了最新的有代表性的沪深股市周收益和日收益数据进行了实证分析,数据从沪深股市成立至今直至2010年7月1日,相比已有研究数据更多、更丰富。同时本文改变了以往时间序列分析采用Eviews软件的缺陷,采用功能更加强大的Matlab软件,得出的结论更加符合实际。
由于分形理论的可预测性,本文推导了分形布朗运动下的Black-Scholes模型,并应用于对股票价格的预测。最后本文分析了沪深股市非线性及分形特征的成因,并对分形理论的进一步研究做出了展望。
Affected by many factors, Stock market price has uncertainty. People are always trying to find a theory to eliminate this uncertainty, to achieve the purpose of forecasting stock prices. From Fama introduced the Efficient Market Theory (EMH), which quickly becomes the cornerstone of modern financial economic, this linear paradigm dominates financial economics for half a century.
However, with the stock market developping, and people raising the awareness of the stock market,as the cornerstone of mordern financial economic,the EMH is proved incompatible with the reality more and more,such as the first month phenomenon and the small company phenomenon. Therefore, many scholars tried to find a better theory to explain these phenomena.The Fractal Theory,which originally used to study the fractal coastline, has been introduced to the study of stock market. The Fractal Theory,which established on the basis of the nonlinear dynamics, relaxes the conditions of the efficient market hypothesis,gives a better explanation of the phenomenon that can not be explained by the efficient market hypothesis. Using the Fractal theory and method to explain the nonlinear characteristics of the Chinese stock market, conducive to understanding and awareness the effectiveness of the stock market, volatility and long-term memory, So it has very important theoretical and practical significance.
This artical selected the Shanghai and Shenzhen Composites as the object, selecting the daily closing price data and the weekly closing price data from the inception to July 1st,2010.using the Eviews and Matlab software and the classic R/S method,analysized the Shanghai and Shenzhen stock market returns,and calculated the H index and the cycle length of the Shanghai and Shenzhen stock market.The results show that:they are Fractal,and their price changes have trends and cyclical. The innovation of this paper is that,tfirst we summary the previous research results,which including the efficient market theory,the concept of fractal market theory and the application. Then we select the latest representative of the Shanghai and Shenzhen stock market returns and weekly earnings data do analysis,which from Shanghai and Shenzhen stock markets'establishment to July 1st,2010.Compared to the data have been,it's more richer. At the same time,we using the Matlab software to avoid the defacts of the Eviews,so the conclusion is more powerful and realistic.
Because of its predictability, this artical derived the Black-Scholes model under Fractional Brownian motion, and applied to the prediction of stock price. Finally, we also analysized the causes of the nonlinear and the fractal,and make a further research prospect of the Fractal Theory.
引文
①引自赵宇新《沪深股市分形特征分析》[J]北京交通大学硕士学位论文.2009.6：5.
②引自埃德加·E·彼得斯《资本市场的混沌与秩序》[M]王小东译.经济科学出版社,1999.3：10.
③引自中国学术论文网“分形市场假说及其在中国证券市场的应用”2009.5.http://www.59168.net/cn/newsshow.asp?id=181144
④转引自朱晓华等《海岸线分维时序动态变化及其分形模拟研究》[J]海洋通报.2002(8)：37-43.
⑤Greene和Fielitz(1977)第一个利用R/S分析法对纽约证券交易所的股票进行了实证研究,发现许多股票的日收益具有长期相关性。
⑥转自Henry O T.Long Memory in Stock Returns:Some International Evidence[J].Applied Finanacial Economics,2002(12):725-729.
⑦转引自Wright J H.Long memory in Emerging Market Stock Returns[J].Emerging Markets Quarterly.2001(5):50-55.
⑧参考自徐龙炳.陆蓉《有效市场理论的前沿研究》[J].财经研究.2001(8)：27-34.
⑨参考自陈秋雨,周生春《分形市场理论的产生和演变评述》[J].社会科学家.2010(11)：51-54.
⑩同上.
11来源于樊智,张世英《金融市场的效率与分形市场理论》[J].系统工程理论与实践.2003(3)：13-19
12园括号中的数字为P值,JB统计量在5%和1%的显著性水平下的临界值分别为5.9915和9.2103.
[1]Fama E.F., Efficient capital markers:A review of theory and empirical work[J], Journal of Finance,1970(25):383-417.
[2]Samuelson. P, Proof that properly anticipated pricefluctuat randomly[J], Industrial Management Reviews,1965,6:41-49.
[3]Mandelbrot B.B., New methods in statistical economics[J],Journal of political economy,1963(71):421-440.
[4]Fama E.F., The behavior of stock market prices[J],Joumal of business,1965,38.
[5]Peters. E, Fractal market analysis:appling chaos theory to investment and economics,1994,New York:John Wiley and Sons.
[6]Lo, A W. Long-term memory in stock market prices [J], Econometrica,1991,5 9(5):1279-1313.
[7]Cheung Y W,Lai K S.A Search for Long Memory in International Stock Market Returns[J] Journal of International Money and Finance,1995(14):597-615.
[8]Cheung Y W.Long Memory in Foreign Exchange Rates[J].Journal of Business and Economics Statistics,1993(11):93-101.
[9]Crato N,Ray B K.Model Selection and Forecasting for Longrange Dependent Processes[J].Journal of Forescating,1996(15):107-125.
[10]R.Mantegna,L.E.Stanley,1998, " Physics Investigation of Financial Markets",In:F.Mallamace,L.E.Stanley(Eds.),Proceedings of the International School of Physics.IOS Press,Amsterdam,473-489.
[11]Eugene F. Fama,R. Roll "Some Properties of Stable Symmetric Distributions"[J] Journal of the American Statistical Association,1971.
[12]B.B.Mandelbrot and A.Blumen.Fractal Geometry:What is it,and What Does it do?[M]Fractals in the Natural Sciences,1989.5.8:3-16.
[13]B.B.Mandelbrot, "Is There Persistence in Stock Price Movements",Seminar on the Analysis of Security Price,University of Chicago,Graduate School of Business,1966.11.12-13.
[14]Greene M T,Fielitz B D.Long-term Dependence in Common Stock Returns[J].Journal of Financial Economics,1977(4):339-349.
[15]Granger C W J,Joyeux R.An Introduction to Long Memory Time Models and Fraction Differencing[J].Journal of Time Series Analysis,1980(1):15-29.
[16]Hosking J R M.Modelling Persistence in Hydrologic Time Series Using Fractional Difference[J].Water Resources Research,1984b(20):1898-1908.
[17]Geweke J,Porter-Hudak S.The Estimation and Application of Long Memory Time Series Models[J].Journal of Time Series Analysis,1983(4):221-238.
[18]Crato N.Some International Evidence Regarding the Stochastic Behavior of Stock Returns[J].Applied Financial Economics,1994(4):33-39.
[19]Wright J H.The Local Asymptotic Power of Certain Tests for Fractional Integration[J].Econometric Theory,1999(15):704-709.
[20]Baillie R T,Chung C F,Tieslau M A.Analyzing Inflation by the Fractionally Integrated ARFIMA-GARCH Model[J].Journal of Applied Econometrics,1996(11):23-40.
[21]Bollerslev T.,Mikkelsen H.Modelling and Pricing Long-memory in Stock Market Volatility[J].Journal of Econometrics,1996(73):151-184.
[22]Calvet L,Fisher A,Mandelbrot B B.Large Deviations and the Distribution of Price Changes[J].Yale University,Working Paper,1997.
[23]B.B.Mandelbrot,1997, "A Multifractal Model of Asset Returns",Paper Provided by Cowles Foundation,Tale University in its series Cowles Foundation Discussion Papers with Number 1164.
[24]T.Lux,and M.Marchesi,1999, "Scaling and Criticality in a Stochastic Multi-agent Model of a Financial Market",Nature 397,498-500.
[25]史永东.上海证券市场的分形结构[J].预测,2000(5)：78-81.
[26]徐龙炳,陆蓉.有效市场理论的前沿研究[J].财经研究,2001(8)：27-34.
[27]张维,黄兴.沪深股市的R/S实证分析[J].系统工程,2001(1)：1-5.
[28]伍海华,李道叶.非线性动力学分析在股票市场中应用-以上海股票市场为例[J].青岛大学学报,2001(12)：5-11.
[29]戴菊贵.有效市场假说与分形市场假说之争[J].金融教学与研究,2003(2)：14-24.
[30]徐绪松,马莉莉,陈彦斌.R/S分析的理论基础：分数布朗运动[J].武汉大学学报(理学版),2004(10)：547-550.
[31]黄诒蓉.中国股市分形结构[J].现代管理科学,2005(2)：53-55.
[32]胡彦梅,张卫国,陈建忠.中国股市长记忆的修正R/S分析[J]数理统计与管理,2006(1)：73-77.
[33]张晓莉,严广乐.中国股票市场长期记忆特征的实证研究[J].系统工程学报,2007(4)：190-194.
[34]王德河.股票价格时间序列分形特征的实证研究[J].审计与经济研究,2007(11)：71-76.
[35]兰若蒙,魏铼,牛敏.R/S分析与中国股市的分形特征[J].哈尔滨商业大学学报(社会科学版),2008(5)：26-30.
[36]王鹏,王建琼.中国股票市场的多分形波动率测度及其有效性研究[J].中国管理科学,2008(12)：9-15.
[37]刘倩,梁久祯.基于R/S方法的股票平均循环周期研究[J].计算机工程与设计,2009,30(21)：4942-4944.
[38]李存金,侯楠楠.基于R/S分析的上海股市有效性实证研究[J].技术经济,2009(5)：71-75.
[39]杨桂元,赵宏宝.中国股市收益率和波动率的长记忆性检验[J].2009(6)：39-43.
[40]苑莹,庄新田.中国股票市场的长记忆性与市场发展状态[J].数理统计与管理,2008(1)：156-163.
[41]胡雪明,宋学锋.深沪股票市场的多重分形分析[J].数量经济技术经济研究,2003(8)：124-127.
[42]施锡铨,艾克凤.股票市场风险的多重分形分析.[J]统计研究,2004(9)：33-36.
[43]曹广喜,史安娜.沪深股市波动的多重分形结构分析[J].经济经纬,2006(6)：136-139.
②引自埃德加·E·彼得斯《资本市场的混沌与秩序》[M]王小东译.经济科学出版社,1999.3：10.
③引自中国学术论文网“分形市场假说及其在中国证券市场的应用”2009.5.http://www.59168.net/cn/newsshow.asp?id=181144
④转引自朱晓华等《海岸线分维时序动态变化及其分形模拟研究》[J]海洋通报.2002(8)：37-43.
⑤Greene和Fielitz(1977)第一个利用R/S分析法对纽约证券交易所的股票进行了实证研究,发现许多股票的日收益具有长期相关性。
⑥转自Henry O T.Long Memory in Stock Returns:Some International Evidence[J].Applied Finanacial Economics,2002(12):725-729.
⑦转引自Wright J H.Long memory in Emerging Market Stock Returns[J].Emerging Markets Quarterly.2001(5):50-55.
⑧参考自徐龙炳.陆蓉《有效市场理论的前沿研究》[J].财经研究.2001(8)：27-34.
⑨参考自陈秋雨,周生春《分形市场理论的产生和演变评述》[J].社会科学家.2010(11)：51-54.
⑩同上.
11来源于樊智,张世英《金融市场的效率与分形市场理论》[J].系统工程理论与实践.2003(3)：13-19
12园括号中的数字为P值,JB统计量在5%和1%的显著性水平下的临界值分别为5.9915和9.2103.
[1]Fama E.F., Efficient capital markers:A review of theory and empirical work[J], Journal of Finance,1970(25):383-417.
[2]Samuelson. P, Proof that properly anticipated pricefluctuat randomly[J], Industrial Management Reviews,1965,6:41-49.
[3]Mandelbrot B.B., New methods in statistical economics[J],Journal of political economy,1963(71):421-440.
[4]Fama E.F., The behavior of stock market prices[J],Joumal of business,1965,38.
[5]Peters. E, Fractal market analysis:appling chaos theory to investment and economics,1994,New York:John Wiley and Sons.
[6]Lo, A W. Long-term memory in stock market prices [J], Econometrica,1991,5 9(5):1279-1313.
[7]Cheung Y W,Lai K S.A Search for Long Memory in International Stock Market Returns[J] Journal of International Money and Finance,1995(14):597-615.
[8]Cheung Y W.Long Memory in Foreign Exchange Rates[J].Journal of Business and Economics Statistics,1993(11):93-101.
[9]Crato N,Ray B K.Model Selection and Forecasting for Longrange Dependent Processes[J].Journal of Forescating,1996(15):107-125.
[10]R.Mantegna,L.E.Stanley,1998, " Physics Investigation of Financial Markets",In:F.Mallamace,L.E.Stanley(Eds.),Proceedings of the International School of Physics.IOS Press,Amsterdam,473-489.
[11]Eugene F. Fama,R. Roll "Some Properties of Stable Symmetric Distributions"[J] Journal of the American Statistical Association,1971.
[12]B.B.Mandelbrot and A.Blumen.Fractal Geometry:What is it,and What Does it do?[M]Fractals in the Natural Sciences,1989.5.8:3-16.
[13]B.B.Mandelbrot, "Is There Persistence in Stock Price Movements",Seminar on the Analysis of Security Price,University of Chicago,Graduate School of Business,1966.11.12-13.
[14]Greene M T,Fielitz B D.Long-term Dependence in Common Stock Returns[J].Journal of Financial Economics,1977(4):339-349.
[15]Granger C W J,Joyeux R.An Introduction to Long Memory Time Models and Fraction Differencing[J].Journal of Time Series Analysis,1980(1):15-29.
[16]Hosking J R M.Modelling Persistence in Hydrologic Time Series Using Fractional Difference[J].Water Resources Research,1984b(20):1898-1908.
[17]Geweke J,Porter-Hudak S.The Estimation and Application of Long Memory Time Series Models[J].Journal of Time Series Analysis,1983(4):221-238.
[18]Crato N.Some International Evidence Regarding the Stochastic Behavior of Stock Returns[J].Applied Financial Economics,1994(4):33-39.
[19]Wright J H.The Local Asymptotic Power of Certain Tests for Fractional Integration[J].Econometric Theory,1999(15):704-709.
[20]Baillie R T,Chung C F,Tieslau M A.Analyzing Inflation by the Fractionally Integrated ARFIMA-GARCH Model[J].Journal of Applied Econometrics,1996(11):23-40.
[21]Bollerslev T.,Mikkelsen H.Modelling and Pricing Long-memory in Stock Market Volatility[J].Journal of Econometrics,1996(73):151-184.
[22]Calvet L,Fisher A,Mandelbrot B B.Large Deviations and the Distribution of Price Changes[J].Yale University,Working Paper,1997.
[23]B.B.Mandelbrot,1997, "A Multifractal Model of Asset Returns",Paper Provided by Cowles Foundation,Tale University in its series Cowles Foundation Discussion Papers with Number 1164.
[24]T.Lux,and M.Marchesi,1999, "Scaling and Criticality in a Stochastic Multi-agent Model of a Financial Market",Nature 397,498-500.
[25]史永东.上海证券市场的分形结构[J].预测,2000(5)：78-81.
[26]徐龙炳,陆蓉.有效市场理论的前沿研究[J].财经研究,2001(8)：27-34.
[27]张维,黄兴.沪深股市的R/S实证分析[J].系统工程,2001(1)：1-5.
[28]伍海华,李道叶.非线性动力学分析在股票市场中应用-以上海股票市场为例[J].青岛大学学报,2001(12)：5-11.
[29]戴菊贵.有效市场假说与分形市场假说之争[J].金融教学与研究,2003(2)：14-24.
[30]徐绪松,马莉莉,陈彦斌.R/S分析的理论基础：分数布朗运动[J].武汉大学学报(理学版),2004(10)：547-550.
[31]黄诒蓉.中国股市分形结构[J].现代管理科学,2005(2)：53-55.
[32]胡彦梅,张卫国,陈建忠.中国股市长记忆的修正R/S分析[J]数理统计与管理,2006(1)：73-77.
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