中国股市高频截面数据的概率分布及其时间相关性研究
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摘要
本文以2000年1月1日至2008年3月31日在沪市交易的所有880只A股股票的1分钟收盘数据为研究对象,引入截面收益率的概念,对两个主要变量——1分钟股票价格截面收益率及其波动率的概率分布、长期记忆性和多重分形特征进行研究。实证研究表明,1分钟股票价格截面收益率服从T分布,特征指数ζ3.1538；1分钟截面收益波动率存在“L”形日内效应,它的概率分布与互补累计概率分布均服从幂律分布,但幂指数不符合负三次方定律。通过降趋脉动分析法对1分钟股票价格收益率及其波动率的时间相关性进行研究,发现1分钟截面收益率的赫斯特指数值接近0.5,呈随机游走状态,不具有长期记忆性；而1分钟截面收益波动率的赫斯特指数接近于1,具有很强的长期记忆性。本文还运用多重分形降趋脉动分析法和数盒子法分别对1分钟股票价格截面收益率及其波动率的多重分形特征进行研究,发现1分钟截面收益率的多重分形标度指数τ(q)为非线性函数,其奇异性强度指数α≠1,且Δα远大于0,说明1分钟股票价格截面收益率具有多重分形特征；而1分钟截面收益波动率的多重分形标度指数τ(q)为线性函数,且τ(q)=q-1,其奇异性强度指数α≈1,且Δa接近于0,说明1分钟股票价格截面收益波动率不具有多重分形特征。
We study the probability distribution, long memory and multi-fracial properties of two important variables, which are the ensemble return and the ensemble volatility of one minute stock close price, using the high-frequency data of all the A stocks'one minute close price from 2000.01.01 to 2008.03.31 in Shanghai stock market. We find that the probability distribution of one minute ensemble return meet the Student's distribution, with its characteristic exponentζ= 3.1538, obeying the inverse cubic law. And we also find that one minute ensemble volatility has intraday effect with "L" shape intraday pattern, its probability distribution and complementary probability distribution are both meet the power-law distribution, but their exponents don't obey the inverse cubic law. We study the long memory of one minute ensemble return using Detrended fluctuation analysis method with the result show that its Hurst index close to 0.5, which indicate that one minute ensemble return follow random work, and doesn't have long memory. On the other hand, the Hurst index of one minute ensemble volatility is close to 1, which indicates one minute ensemble volatility has long memory. We respectively study the multiracial characterize of one minute ensemble return and its volatility with Multiracial detrended fluctuation analysis method and Box-counting method, and find that the multiracial scaling exponentτ(q) of one minute ensemble return is a nonlinear function, the singularity strength exponentαdoesn't equal to 1, and△αis much larger than 0, which indicate that one minute ensemble return has multiracial characterize; otherwise, the multiracial scaling exponentτ(q) of one minute ensemble volatility is a linear function, its singularity strength exponent a is close to 1, and△αis close to 0, which indicate that one minute ensemble return doesn't have multiracial characterize.
We study the probability distribution, long memory and multi-fracial properties of two important variables, which are the ensemble return and the ensemble volatility of one minute stock close price, using the high-frequency data of all the A stocks'one minute close price from 2000.01.01 to 2008.03.31 in Shanghai stock market. We find that the probability distribution of one minute ensemble return meet the Student's distribution, with its characteristic exponentζ= 3.1538, obeying the inverse cubic law. And we also find that one minute ensemble volatility has intraday effect with "L" shape intraday pattern, its probability distribution and complementary probability distribution are both meet the power-law distribution, but their exponents don't obey the inverse cubic law. We study the long memory of one minute ensemble return using Detrended fluctuation analysis method with the result show that its Hurst index close to 0.5, which indicate that one minute ensemble return follow random work, and doesn't have long memory. On the other hand, the Hurst index of one minute ensemble volatility is close to 1, which indicates one minute ensemble volatility has long memory. We respectively study the multiracial characterize of one minute ensemble return and its volatility with Multiracial detrended fluctuation analysis method and Box-counting method, and find that the multiracial scaling exponentτ(q) of one minute ensemble return is a nonlinear function, the singularity strength exponentαdoesn't equal to 1, and△αis much larger than 0, which indicate that one minute ensemble return has multiracial characterize; otherwise, the multiracial scaling exponentτ(q) of one minute ensemble volatility is a linear function, its singularity strength exponent a is close to 1, and△αis close to 0, which indicate that one minute ensemble return doesn't have multiracial characterize.
引文
[1]Andersen T G., Bollerslev T. Intraday and interday volatility in the Japanese Stock Market. International Finance Markets.2000,10(M):107-130
[2]房振明,王春峰.上海股票市场收益日内效应的研究.北京理工大学学报.2004,6(3) ： 38-41
[3]Bechelier, L. Theory of speculation. Ph. D. thesis. University of Paris.1990
[4]张元萍.数理金融.中国金融出版社.2004,62-65
[5]Mandelbrot B.B. New methods in statistical economics. Journal of Political Economy. 1963,71:421-440
[6]Mandelbrot B.B. The variation of certain speculative prices. Journal of Business,1963, 36:394-419
[7]Fama E. F. The behavior of stock market price. Journal of Business.1965,38(1): 34-105
[8]Fama, E. F. Portfolio analysis in a stable pertain market. Journal of Business.1963,36, 420-429
[9]Mandelbrot B.B. The variation of some other speculative prices. Journal of Business.-1967,40:393-413
[10]柳会珍,顾岚.股票收益率分布的尾部行为研究.系统工程.2005,23(2)：74-77
[11]徐龙炳.中国股票市场股票收益稳态特性的实证研究.金融研究.2001,6：36-43
[12]王建华,王玉玲,柯开明.中国股票收益率的稳定分布拟合与检验.武汉理工大学学报.2003,25(10)：99-102
[13]武东,张青,汤银才.基于稳定分布的股票收益率的特征分析.统计与决策.2006,10：92-93
[14]孙群.中国股票收益率的稳定分布拟合与检验.数学理论与应用.2009,29(2)：56-58
[15]Mantegna R.N., H.E.Stanley. Scaling behavior in the dynamics of an economic index. Nature.1995,376:46-49
[16]Hari M.Gupta, Jose R.Campanha. The Gradually Truncated Levy Flight for Systems with Power-Law Distributions. Physica A.1999,268
[17]M.Yu.Romanovsky. Truncated Lew Distribution of SP500 Stock Index Fluctuations. Distribution of one-share fluctuations in a model space. Physica A.2000,287
[18]L.Couto Mirandaa, b.R.Rieraa. Truncated Lew Walks and an Emerging Market Economic Index. Physica A.2001,297
[19]胡锡健,韩东.考虑成交量的股票收益率的渐进分布.应用概率统计.2008,24(1)：83-97
[20]Gopikrishnan P., M. Meyer, L.A.N. Amaral, H.E. Stanley. Inverse cubic law for the distribution of stock price variations. European Physical Journal.1998, B3:139-140
[21]Kaizoji T., M. Kaizoji. Power law for the ensemble of stock price. Physical A.2004,344: 240-243
[22]Coronel-Brizio H.F., A.R. Hernandez-Montoya. Asymptotic behavior of the daily increment distribution of the IPC, the Mexican stock market index. Revisal Mexicana de Fisica.2005,51:27-31
[23]Lee K.E., J.W. Lee. Scaling Properties of price changes for Korean stock indices. Journal of the Korean Physical Society.2004,44:668-671
[24]W.X. Zhou, G.F. Gu. Statistical properties of daily ensemble variables in the Chinese stock markets. Physica A.2007,383:497-506
[25]徐天群,刘焕彬,徐天河,陈跃鹏.股票收益率的组合分布研究.武汉理工大学学报.2009,31(1)：127-130
[26]G.F. Gu, W. Chen, W.X. Zhou. Empirical distributions of Chinese stock returns at different microscopic timescales. Physica A.2008,387:495-502
[27]胡小平,赵梅,何建敏.涨跌停板股票收益率的概率密度估计及其应用.统计与决策.2008,15：36-39
[28]史道济,高峰.股票收益率的次指数分布拟合.数理统计与管理.2003,22(6)：24-28
[29]Fernandez C., M. Steel. On Bayesian Modelling of Fat Tails and Skewness. Journal of the American Statistical Association.1998,93
[30]崔畅.基于广义误差分布的股票收益率波动特征分析.宁波大学学报.2009,22(4)：94-98
[31]Tim Bollerslev. A Conditionally Heteroskedastic Time Series Model for speculative Prices and Rates of Return. The Review of Economics and Statistics.1987,69(3): 542-547
[32]Andersen T.G., Bollerslev T. Answering the skeptics:Yes, Standard Volatility Models Do Provide Accurate Forecasts. International Economic Review.1998,39:885-905
[33]Goyal A. Predictability of stock return volatility from GARCH Models. Anderson Graduate School of Management, UCLA.2000
[34]Gavala A., Nikolay Gospodinov, Deming Jiang. Forecasting Stock Market Volatility. University of Edinburgh Center for Financial Markets Research Working Paper.2002, 04
[35]罗羡华,李元.股票波动率的高频率数据估计及实证分析.广州大学学报.2003,2(4)：307-310
[36]柳会珍,顾岚.股票收益率波动和极值关系研究.统计研究.2006,10：68-71
[37]周林.股票波动率模拟及预测效果的实证研究.上海经济研究.2006,12：100-105
[38]施红俊,陈伟忠.股票月收益实际波动率的实证研究.同济大学学报.2005,33(2)：264-268
[39]Hurst H. Long term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers.1951,116:770-799
[40]Mandelbrot B B. When Call price be arbitraged efficiently? A limit to validity of the random walk and martingale models. Review of Economies and Statistics.1971,53(2): 225-236
[41]Lo.A W. Long-term memory in stock market prices. Econometrics.1991, 59(6):1279-1313
[42]张维,黄兴。沪深股市的R/S实证分析.系统工程.2001,19(1)：1-5
[43]陈梦根.股票价格分形特征的实证研究：修正R/S分析.统计研究.2003,4：57-61
[44]C.K.Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L.Goldberger. Mosaic organization of DNA nucleotides. Phys.Rev.1994, E(49):1685-1689
[45]M. Taqqu, V. Teverovsky, W. Willinger. Estimators for long-range dependence:an empirical study. Fractals.1995,3:785-798
[46]A. Montanari, M.S. Taqqu, V. Teverovsky. Estimating long-range dependence in the presence of periodicity:an empirical study. Math. Compute. Model.1999,29(10-12): 217-228
[47]B. Audit, E. Bacry, J.-F. Muzy, A. Arne'odo. Wavelet-based estimators of scaling behavior. IEEE Trans. Inf. Theory.2002,48:2938-2954
[48]F.R., G.F. Gu, W.X. Zhou. Scaling and memory in the return intervals of realized volatility. Physica A:Statistical Mechanics and its Applications.2009,22(388): 4787-4796
[49]Giraitis L, Kokoszka P, Leipus R. Rescaled varisnce and related tests for long memory in volatility and levels. Journal of Econometrics.2003,112:265-294
[50]余俊,姜伟,龙琼华.国际股票市场收益率和波动率的长记忆性研究.财贸研究.2007,5：84-90
[51]余俊,方爱丽,熊文海.国际股票市场收益的长记忆性比较研究.中国管理科学.2008,16(4)：24-29
[2]房振明,王春峰.上海股票市场收益日内效应的研究.北京理工大学学报.2004,6(3) ： 38-41
[3]Bechelier, L. Theory of speculation. Ph. D. thesis. University of Paris.1990
[4]张元萍.数理金融.中国金融出版社.2004,62-65
[5]Mandelbrot B.B. New methods in statistical economics. Journal of Political Economy. 1963,71:421-440
[6]Mandelbrot B.B. The variation of certain speculative prices. Journal of Business,1963, 36:394-419
[7]Fama E. F. The behavior of stock market price. Journal of Business.1965,38(1): 34-105
[8]Fama, E. F. Portfolio analysis in a stable pertain market. Journal of Business.1963,36, 420-429
[9]Mandelbrot B.B. The variation of some other speculative prices. Journal of Business.-1967,40:393-413
[10]柳会珍,顾岚.股票收益率分布的尾部行为研究.系统工程.2005,23(2)：74-77
[11]徐龙炳.中国股票市场股票收益稳态特性的实证研究.金融研究.2001,6：36-43
[12]王建华,王玉玲,柯开明.中国股票收益率的稳定分布拟合与检验.武汉理工大学学报.2003,25(10)：99-102
[13]武东,张青,汤银才.基于稳定分布的股票收益率的特征分析.统计与决策.2006,10：92-93
[14]孙群.中国股票收益率的稳定分布拟合与检验.数学理论与应用.2009,29(2)：56-58
[15]Mantegna R.N., H.E.Stanley. Scaling behavior in the dynamics of an economic index. Nature.1995,376:46-49
[16]Hari M.Gupta, Jose R.Campanha. The Gradually Truncated Levy Flight for Systems with Power-Law Distributions. Physica A.1999,268
[17]M.Yu.Romanovsky. Truncated Lew Distribution of SP500 Stock Index Fluctuations. Distribution of one-share fluctuations in a model space. Physica A.2000,287
[18]L.Couto Mirandaa, b.R.Rieraa. Truncated Lew Walks and an Emerging Market Economic Index. Physica A.2001,297
[19]胡锡健,韩东.考虑成交量的股票收益率的渐进分布.应用概率统计.2008,24(1)：83-97
[20]Gopikrishnan P., M. Meyer, L.A.N. Amaral, H.E. Stanley. Inverse cubic law for the distribution of stock price variations. European Physical Journal.1998, B3:139-140
[21]Kaizoji T., M. Kaizoji. Power law for the ensemble of stock price. Physical A.2004,344: 240-243
[22]Coronel-Brizio H.F., A.R. Hernandez-Montoya. Asymptotic behavior of the daily increment distribution of the IPC, the Mexican stock market index. Revisal Mexicana de Fisica.2005,51:27-31
[23]Lee K.E., J.W. Lee. Scaling Properties of price changes for Korean stock indices. Journal of the Korean Physical Society.2004,44:668-671
[24]W.X. Zhou, G.F. Gu. Statistical properties of daily ensemble variables in the Chinese stock markets. Physica A.2007,383:497-506
[25]徐天群,刘焕彬,徐天河,陈跃鹏.股票收益率的组合分布研究.武汉理工大学学报.2009,31(1)：127-130
[26]G.F. Gu, W. Chen, W.X. Zhou. Empirical distributions of Chinese stock returns at different microscopic timescales. Physica A.2008,387:495-502
[27]胡小平,赵梅,何建敏.涨跌停板股票收益率的概率密度估计及其应用.统计与决策.2008,15：36-39
[28]史道济,高峰.股票收益率的次指数分布拟合.数理统计与管理.2003,22(6)：24-28
[29]Fernandez C., M. Steel. On Bayesian Modelling of Fat Tails and Skewness. Journal of the American Statistical Association.1998,93
[30]崔畅.基于广义误差分布的股票收益率波动特征分析.宁波大学学报.2009,22(4)：94-98
[31]Tim Bollerslev. A Conditionally Heteroskedastic Time Series Model for speculative Prices and Rates of Return. The Review of Economics and Statistics.1987,69(3): 542-547
[32]Andersen T.G., Bollerslev T. Answering the skeptics:Yes, Standard Volatility Models Do Provide Accurate Forecasts. International Economic Review.1998,39:885-905
[33]Goyal A. Predictability of stock return volatility from GARCH Models. Anderson Graduate School of Management, UCLA.2000
[34]Gavala A., Nikolay Gospodinov, Deming Jiang. Forecasting Stock Market Volatility. University of Edinburgh Center for Financial Markets Research Working Paper.2002, 04
[35]罗羡华,李元.股票波动率的高频率数据估计及实证分析.广州大学学报.2003,2(4)：307-310
[36]柳会珍,顾岚.股票收益率波动和极值关系研究.统计研究.2006,10：68-71
[37]周林.股票波动率模拟及预测效果的实证研究.上海经济研究.2006,12：100-105
[38]施红俊,陈伟忠.股票月收益实际波动率的实证研究.同济大学学报.2005,33(2)：264-268
[39]Hurst H. Long term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers.1951,116:770-799
[40]Mandelbrot B B. When Call price be arbitraged efficiently? A limit to validity of the random walk and martingale models. Review of Economies and Statistics.1971,53(2): 225-236
[41]Lo.A W. Long-term memory in stock market prices. Econometrics.1991, 59(6):1279-1313
[42]张维,黄兴。沪深股市的R/S实证分析.系统工程.2001,19(1)：1-5
[43]陈梦根.股票价格分形特征的实证研究：修正R/S分析.统计研究.2003,4：57-61
[44]C.K.Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L.Goldberger. Mosaic organization of DNA nucleotides. Phys.Rev.1994, E(49):1685-1689
[45]M. Taqqu, V. Teverovsky, W. Willinger. Estimators for long-range dependence:an empirical study. Fractals.1995,3:785-798
[46]A. Montanari, M.S. Taqqu, V. Teverovsky. Estimating long-range dependence in the presence of periodicity:an empirical study. Math. Compute. Model.1999,29(10-12): 217-228
[47]B. Audit, E. Bacry, J.-F. Muzy, A. Arne'odo. Wavelet-based estimators of scaling behavior. IEEE Trans. Inf. Theory.2002,48:2938-2954
[48]F.R., G.F. Gu, W.X. Zhou. Scaling and memory in the return intervals of realized volatility. Physica A:Statistical Mechanics and its Applications.2009,22(388): 4787-4796
[49]Giraitis L, Kokoszka P, Leipus R. Rescaled varisnce and related tests for long memory in volatility and levels. Journal of Econometrics.2003,112:265-294
[50]余俊,姜伟,龙琼华.国际股票市场收益率和波动率的长记忆性研究.财贸研究.2007,5：84-90
[51]余俊,方爱丽,熊文海.国际股票市场收益的长记忆性比较研究.中国管理科学.2008,16(4)：24-29