中国股票市场Hurst指数与多重分形分析
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摘要
有效市场假说是建立在理性投资者、有效市场及随机游走过程这三个核心前提假定基础上的。有效市场假说成立依赖于市场呈独立正态分布、方差存在的假设,但是现实市场是一个复杂的相互依存的系统。许多情况是有效市场假说无法解释得,例如规模效应、季节效应等。因此,众多的研究人员开始关注一些非线性理论在金融市场的运用,典型的有突变理论、协同市场假定、行为金融理论、分形市场假说等。
分形理论是非线性分析方法中的一种,由于它能有效的解释自然和经济现象中许多及其复杂多变的问题,因此它已经成为非线性资本市场理论的有力分析工具。本文共分为五个部分,第一部分:简要介绍了有效市场假说理论(EMH)及分形市场假说理论(FMH)的发展及主要思想;并列举了国内外的研究现状。第二部分:详细介绍了分形理论的主要参数Hurst指数及多重分形谱的计算方法。第三部分:利用上证指数1991年1月2日至2005年8月31日及深圳成指1991年4月1日至2006年1月8日的日收盘价为样本数据进行分析,并由此得出中国股票市场的分形特性。第四部分:任选上证指数、深圳成指的一段数据进行多重分形分析,并利用盒维数法对上证指数进行多重谱函数分析,得出沪深股市皆具有多重分形特征。第五部分:结论及展望。
通过对上证指数日收盘价的研究发现,沪市存在明显的多重分形特征,但仅用传统的打乱顺序计算Hurst指数无法证明其分形性,必须与V统计量结合使用;通过实证研究得到了上证指数存在一个300天的非循环周期,H值为0.626358,深圳成指存在一个为期1200天的非循环周期,H值为0.590986;利用盒维数法计算多重分形谱函数证明了股票市场的多重分形性。
EMH is necessary to prove that price follows a random walk model. But EMH is established on market, which is independent normal distribution and variance existing. But the market is a complicated interdependent system. A lot of cases are what EMH can’t explain, such as scale effect, season effect etc. So a lot of theories that explain these special cases have appeared, such as catastrophe theory, behavior finance theory, coherent market theory and the fractal market theory, which let researchers taking more attention more about the use of non-linear theory in financial market.
Fractal theory is one kind of nonlinear analysis and has only decades since it developed, because it can efficiently expenses many complicated problem, such as economics phenomena, so it has became the strong analysis tool of the nonlinear capital market theory.
There are five parts in our paper, the first is introduced simply the theories of EMH and FMH; the second is introduced particularly the method of Hurst index and multifractal; and in the third we get the conclusion that China stock market is FMH with analysis using the date of Shanghai from 2 Jan. 1991 to 31 Aug. 2005 and Shenzhen from 1 Apr. 1991 to 8 Jan. 2006; in the forth we get the conclusion that China stock market is MFMH with analysis the date of Shanghai and Shenzhen. The fifth is conclusion and expectation.
In our paper, researching on the index of shanghai and Shenzhen bond market, we find that our bond market has obvious fractal character, and working out the Hurst index and V index, which prove that our bond market can be test by non-linear theory. We can get the 300 days as a period, and the H is 0.626358 of shanghai bond market; the same as 1200 days and H is 0.590986 of Shenzhen bond market. In the last we use Box-counting method to prove that Shanghai and Shenzhen bond market is mutifractal.
分形理论是非线性分析方法中的一种,由于它能有效的解释自然和经济现象中许多及其复杂多变的问题,因此它已经成为非线性资本市场理论的有力分析工具。本文共分为五个部分,第一部分:简要介绍了有效市场假说理论(EMH)及分形市场假说理论(FMH)的发展及主要思想;并列举了国内外的研究现状。第二部分:详细介绍了分形理论的主要参数Hurst指数及多重分形谱的计算方法。第三部分:利用上证指数1991年1月2日至2005年8月31日及深圳成指1991年4月1日至2006年1月8日的日收盘价为样本数据进行分析,并由此得出中国股票市场的分形特性。第四部分:任选上证指数、深圳成指的一段数据进行多重分形分析,并利用盒维数法对上证指数进行多重谱函数分析,得出沪深股市皆具有多重分形特征。第五部分:结论及展望。
通过对上证指数日收盘价的研究发现,沪市存在明显的多重分形特征,但仅用传统的打乱顺序计算Hurst指数无法证明其分形性,必须与V统计量结合使用;通过实证研究得到了上证指数存在一个300天的非循环周期,H值为0.626358,深圳成指存在一个为期1200天的非循环周期,H值为0.590986;利用盒维数法计算多重分形谱函数证明了股票市场的多重分形性。
EMH is necessary to prove that price follows a random walk model. But EMH is established on market, which is independent normal distribution and variance existing. But the market is a complicated interdependent system. A lot of cases are what EMH can’t explain, such as scale effect, season effect etc. So a lot of theories that explain these special cases have appeared, such as catastrophe theory, behavior finance theory, coherent market theory and the fractal market theory, which let researchers taking more attention more about the use of non-linear theory in financial market.
Fractal theory is one kind of nonlinear analysis and has only decades since it developed, because it can efficiently expenses many complicated problem, such as economics phenomena, so it has became the strong analysis tool of the nonlinear capital market theory.
There are five parts in our paper, the first is introduced simply the theories of EMH and FMH; the second is introduced particularly the method of Hurst index and multifractal; and in the third we get the conclusion that China stock market is FMH with analysis using the date of Shanghai from 2 Jan. 1991 to 31 Aug. 2005 and Shenzhen from 1 Apr. 1991 to 8 Jan. 2006; in the forth we get the conclusion that China stock market is MFMH with analysis the date of Shanghai and Shenzhen. The fifth is conclusion and expectation.
In our paper, researching on the index of shanghai and Shenzhen bond market, we find that our bond market has obvious fractal character, and working out the Hurst index and V index, which prove that our bond market can be test by non-linear theory. We can get the 300 days as a period, and the H is 0.626358 of shanghai bond market; the same as 1200 days and H is 0.590986 of Shenzhen bond market. In the last we use Box-counting method to prove that Shanghai and Shenzhen bond market is mutifractal.
引文
[1] 高鸿桢,关于上海股市效率性的研究,厦门大学学报,1996(4)
[2] 陈旭,刘勇,对我国股票市场有效性的实证分析及政策建议,证券投资,1999(3)
[3] 张兵,中国股票市场有效性研究,南京大学出版社,2004
[4] 丁玲,蒋毅,资本市场中分形市场的分析探讨,江苏大学学报,2002(3)
[5] 曹宏铎,经济系统分形机制与股票市场 R/S 分析,系统工程理论与实践,2003(3)
[6] 陈千里,股市波动的周内效应研究,湖北大学学报(自科版),2003,(3):29-32
[7] 樊智,张世英,金融市场的效率与分形市场理论,系统工程理论与实践,2002(3)
[8] 林勇,郭林军,有效市场假设与分形市场假设,预测,2002(2)
[9] Fama.E.F. The behavior of Stock Market Prices. Journal of Business, 38
[10] Fama.F.E. Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance, 1970, 25:383~417
[11] 俞乔,市场有效、周期异常与股价波动,经济研究,1994(9)
[12] 陈梦根,中国股市长期记忆效应的实证研究,经济研究,2003(3)
[13] 陈小悦,姚怡,上海股市风险与收益定量分析,经济科学,1995(1)
[14] 易丹辉,数据分析与 EViews 应用,中国统计出版社,2002,10
[15] 张晓峒,计量经济学软件 Eviews 使用指南,南开大学出版社,2003
[16] Ghashghaie S, Breymann W, Peinke J, Talkner T, Dodge Y. Turbulent cascade in foreign exchange markets. Nature,1996,381:767~770
[17] Kendall.M. The Analysis of Economic time series: part1 Prices. Journal of Royal Statistical Society, 1953
[18] Anderson.M.K. Do long-memory models have long memory. International Journal of Forecasting,2000,16:121~124
[19] Andrew W.Lo. Long-Term memory in Stock Market Price. Econometrics, 1991
[20] B.Mandelbrot,J.R, Wallis. Robustess of the Rescaled rang R/S in the measurement of non-cyclic long run statistical dependence, Water Resoures, 1969
[21] Mandelbrot B B, Passoja D E, Paullay A J. Fractal character of fracture surface of metals. Nature, 1984, 308 (19): 721~723
[22] Mandelbrot B B. A Multifractal Walk Down Wall Street. Scientific American, 1999, 5: 20~23.
[23] Mandelbrot.B.B. Fractals and scaling in finance. Springer, 1997 .371~419
[24] Edgar.E.Peters. Chaos and order in the capital markers.2rd ,1996
[25] Edgar.E.Peters. Fractal markets analysis-applying chaos theory to investment and economics, 1996
[26] 宋颂兴,封闭型证券投资基金定价研究,数量经济技术经济研究,1999(9)
[27] 吴世农,我国证券市场效率的分析,经济研究,1996(4)
[28] 徐绪讼,陈彦斌,深沪股市分形维实证研究,数量经济技术经济研究,2000(10)
[29] 徐科军,资本市场的混沌与分形-从有效市场假说到分形市场假说,河南金融管理干部学院学报,2002(3)
[30] 熊正丰,金融时间序列分形维估计的小波方法,系统工程理论与实践,2002(12)
[31] 王春峰,蒋祥林,李刚,基于随即波动性模型的中国股市波动性估计,管理科学学报,2003(4)
[32] 王春峰,张庆翠,李刚,中国股票市场收益的长期记忆性研究,系统工程,2003(1)
[33] Arduini F, Fioravanti S, Giusto D D. A multifractal-based approach to natural scene analysis,CCECE’ 91, 1991:2681~2684
[34] Brown G, Michon G, Peyriere J. On the multifractal analysis of measures. Journal of Stat Phys.,1992, 66 :775~790
[35] Halsey T C, Jensen M H, Kadanoff L P. Fractal measures and their singularities: the characterization of strange sets. Phys Rev A, 1986, 33:1141~1150
[36] Rosario N, Mantegna, H Eugene Stanley, Modeling of financial data: Comparison of the truncated levy flight and the ARCH(1) and GARCH(1,1) processes. phsica A,1998,254:77~84
[37] Ioannis Andreadis, Apostolos Serletis. Evidence of a random multifractal turbulent structure in the Dow Jones Industrial Average. Chaos, Solitons and Fractals, 2002, 13: 1309~1315.
[38] Fulco U L, Lyra M L, Petroni F, Serva M,Viswanathan G M. Astochastic model for multifractal behavior of stock price. International Journal of modern physics B, 2004,18(4~5):681~689
[39] 张永东,毕秋香,中国股票市场多标度行为的实证分析,预测,2002(4):56~59
[40] 何建敏,常松,中国股票市场多重分形游走及其预测,中国管理科学, 2002(3): 11~17
[41] 胡雪明,宋学锋,深沪股票市场的多重分形分析, 数量经济技术经济研究, 2003(8):124~127
[42] Ding Shun Ho, Chung Kung Lee, Cheng Cai Wang, Mang Chuang. Scaling characteristics in the Taiwan stock market, Physica A, 2004, 332: 448-460.
[43] 施锡铨, 艾克凤,股票市场风险的多重分形分析, 统计研究,2004, 9:33~36
[44] 周孝华,宋坤,高频金融时间序列的异象特征分析及应用-基于多重分形谱及其参数的研究,财经研究,2005(7)
[45] Hurst H.E. Long-term storage of reservoirs. Trans. Amer. Soc. Civil. Eng, 1995, 116:770-808.
[46] 陈春晖,中国股票市场有效性的统计研究,[硕士论文],湖南大学,2004,11
[47] 卢方元,中国股票收益率的多重分形分析,系统工程理论与实践,2003(8):124-127
[48] 卢方元,中国股市收益率波动研究,[博士论文],西南交通大学,2005,5
[49] Xia Sun, Huiping Chen, Ziqin Wu, Yongzhuang Yuan.Predictability of multifractal analysis of Hang Seng index in Hong Kong stock market. Physica A, 2001, 291: 553-562.
[50] Huiping Chen, Xia Sun, Ziqin Wu,Binghong Wang. Enlightenment from various conditional probabilities about Hang Seng index in Hong Kong stock market. Physica A, 2004, 335: 183-196.
[51] Multifractal analysis of Hang Seng index in Hong Kong stock market. Physica A, 2001, 291: 553-562.
[52] 孙霞,傅竹西,吴自勤,薄膜生长的多重分形谱的计算,计算物理,2001(3):247~252
[53] 何光辉,经济时间序列的统计方法,世界经济研究,2004,(2):50-53
[2] 陈旭,刘勇,对我国股票市场有效性的实证分析及政策建议,证券投资,1999(3)
[3] 张兵,中国股票市场有效性研究,南京大学出版社,2004
[4] 丁玲,蒋毅,资本市场中分形市场的分析探讨,江苏大学学报,2002(3)
[5] 曹宏铎,经济系统分形机制与股票市场 R/S 分析,系统工程理论与实践,2003(3)
[6] 陈千里,股市波动的周内效应研究,湖北大学学报(自科版),2003,(3):29-32
[7] 樊智,张世英,金融市场的效率与分形市场理论,系统工程理论与实践,2002(3)
[8] 林勇,郭林军,有效市场假设与分形市场假设,预测,2002(2)
[9] Fama.E.F. The behavior of Stock Market Prices. Journal of Business, 38
[10] Fama.F.E. Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance, 1970, 25:383~417
[11] 俞乔,市场有效、周期异常与股价波动,经济研究,1994(9)
[12] 陈梦根,中国股市长期记忆效应的实证研究,经济研究,2003(3)
[13] 陈小悦,姚怡,上海股市风险与收益定量分析,经济科学,1995(1)
[14] 易丹辉,数据分析与 EViews 应用,中国统计出版社,2002,10
[15] 张晓峒,计量经济学软件 Eviews 使用指南,南开大学出版社,2003
[16] Ghashghaie S, Breymann W, Peinke J, Talkner T, Dodge Y. Turbulent cascade in foreign exchange markets. Nature,1996,381:767~770
[17] Kendall.M. The Analysis of Economic time series: part1 Prices. Journal of Royal Statistical Society, 1953
[18] Anderson.M.K. Do long-memory models have long memory. International Journal of Forecasting,2000,16:121~124
[19] Andrew W.Lo. Long-Term memory in Stock Market Price. Econometrics, 1991
[20] B.Mandelbrot,J.R, Wallis. Robustess of the Rescaled rang R/S in the measurement of non-cyclic long run statistical dependence, Water Resoures, 1969
[21] Mandelbrot B B, Passoja D E, Paullay A J. Fractal character of fracture surface of metals. Nature, 1984, 308 (19): 721~723
[22] Mandelbrot B B. A Multifractal Walk Down Wall Street. Scientific American, 1999, 5: 20~23.
[23] Mandelbrot.B.B. Fractals and scaling in finance. Springer, 1997 .371~419
[24] Edgar.E.Peters. Chaos and order in the capital markers.2rd ,1996
[25] Edgar.E.Peters. Fractal markets analysis-applying chaos theory to investment and economics, 1996
[26] 宋颂兴,封闭型证券投资基金定价研究,数量经济技术经济研究,1999(9)
[27] 吴世农,我国证券市场效率的分析,经济研究,1996(4)
[28] 徐绪讼,陈彦斌,深沪股市分形维实证研究,数量经济技术经济研究,2000(10)
[29] 徐科军,资本市场的混沌与分形-从有效市场假说到分形市场假说,河南金融管理干部学院学报,2002(3)
[30] 熊正丰,金融时间序列分形维估计的小波方法,系统工程理论与实践,2002(12)
[31] 王春峰,蒋祥林,李刚,基于随即波动性模型的中国股市波动性估计,管理科学学报,2003(4)
[32] 王春峰,张庆翠,李刚,中国股票市场收益的长期记忆性研究,系统工程,2003(1)
[33] Arduini F, Fioravanti S, Giusto D D. A multifractal-based approach to natural scene analysis,CCECE’ 91, 1991:2681~2684
[34] Brown G, Michon G, Peyriere J. On the multifractal analysis of measures. Journal of Stat Phys.,1992, 66 :775~790
[35] Halsey T C, Jensen M H, Kadanoff L P. Fractal measures and their singularities: the characterization of strange sets. Phys Rev A, 1986, 33:1141~1150
[36] Rosario N, Mantegna, H Eugene Stanley, Modeling of financial data: Comparison of the truncated levy flight and the ARCH(1) and GARCH(1,1) processes. phsica A,1998,254:77~84
[37] Ioannis Andreadis, Apostolos Serletis. Evidence of a random multifractal turbulent structure in the Dow Jones Industrial Average. Chaos, Solitons and Fractals, 2002, 13: 1309~1315.
[38] Fulco U L, Lyra M L, Petroni F, Serva M,Viswanathan G M. Astochastic model for multifractal behavior of stock price. International Journal of modern physics B, 2004,18(4~5):681~689
[39] 张永东,毕秋香,中国股票市场多标度行为的实证分析,预测,2002(4):56~59
[40] 何建敏,常松,中国股票市场多重分形游走及其预测,中国管理科学, 2002(3): 11~17
[41] 胡雪明,宋学锋,深沪股票市场的多重分形分析, 数量经济技术经济研究, 2003(8):124~127
[42] Ding Shun Ho, Chung Kung Lee, Cheng Cai Wang, Mang Chuang. Scaling characteristics in the Taiwan stock market, Physica A, 2004, 332: 448-460.
[43] 施锡铨, 艾克凤,股票市场风险的多重分形分析, 统计研究,2004, 9:33~36
[44] 周孝华,宋坤,高频金融时间序列的异象特征分析及应用-基于多重分形谱及其参数的研究,财经研究,2005(7)
[45] Hurst H.E. Long-term storage of reservoirs. Trans. Amer. Soc. Civil. Eng, 1995, 116:770-808.
[46] 陈春晖,中国股票市场有效性的统计研究,[硕士论文],湖南大学,2004,11
[47] 卢方元,中国股票收益率的多重分形分析,系统工程理论与实践,2003(8):124-127
[48] 卢方元,中国股市收益率波动研究,[博士论文],西南交通大学,2005,5
[49] Xia Sun, Huiping Chen, Ziqin Wu, Yongzhuang Yuan.Predictability of multifractal analysis of Hang Seng index in Hong Kong stock market. Physica A, 2001, 291: 553-562.
[50] Huiping Chen, Xia Sun, Ziqin Wu,Binghong Wang. Enlightenment from various conditional probabilities about Hang Seng index in Hong Kong stock market. Physica A, 2004, 335: 183-196.
[51] Multifractal analysis of Hang Seng index in Hong Kong stock market. Physica A, 2001, 291: 553-562.
[52] 孙霞,傅竹西,吴自勤,薄膜生长的多重分形谱的计算,计算物理,2001(3):247~252
[53] 何光辉,经济时间序列的统计方法,世界经济研究,2004,(2):50-53
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