股票收益分布函数分析及价格预测
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摘要
我们研究股票市场价格时，通常认为股票价格模型服从布朗运动，即对数收益率是正态分布的。许多以股票为标的物的金融衍生产品就是以此模型作为定价的基础。然而对实际市场数据的经验统计结果表明，多数股票的对数收益率并不服从正态分布。
本文首先讨论了股票收益分布的种类，作者对前人的工作综合发现目前描述股票收益的函数大概有七种：高斯正态分布、利维稳定分布、t标度分布、尖峰态分布、随机波动率模型、ARCH—GARCH模型、分形布朗运动，我们分别对这七种函数进行了简要的介绍。
然后对股票收益是否服从正态分布进行检验，采用偏度、峰度检验和x~2检验对上海股票市场的几支股票进行了检验，发现以天为时间尺度，股票收益并不服从正态分布。接着我们对股票收益函数进行拟合，第一种方法是采用Mantegna和Stanley在1995年研究美国纽约股票交易所S&P500指数的收益分布所采用的方法，对上证综合指数进行了拟合。发现利维稳定分布能很好的描述上证综合指数。第二种方法是对Laplace分布和正态分布进行对比，看Laplace分布是否比正态分布更优越。结果发现在我们研究的股票中，所有股票的日收益都不服从正态分布，而大多数股票收益服从Laplace分布。但是随着时间间距的放大，服从正态分布的股票逐渐增多，这与Fama关于长期收益服从正态分布的假设是一致的。我们又从股票市场上的投资者进行入手提出了一个简单的股票价格模型，并利用此模型对股票收益函数进行分析。
最后，我接着建立了两种模型对股票价格进行预测，第一种是灰色—马尔柯夫模型，发现其相对于灰色理论模型来说精确度还是比较高的；第二种是经过修正的时间序列模型。
When we study stock market, we usually think that the model of stock price obeys Brownian movement, which log-return characterize normal distribution. It is the foundation that we define a lot of financial derivative which take stock as thing marked. But Statistics result of actual market indicate most log-return of stock disobey normal distribution.
In the article we at first discuss the kind of stock log-return distribution. We find that 7 kinds function can describe stock log-return form their works: Gausses of normal distribution, Levy distribution, t distribution, spike attitude distribution, random fluctuating model, ARCH-GARCH model, divide shape Brownian movement .We have introduced brief these seven kinds of function.
Then we had examined several stocks of the stock market of Shanghai with Partial degree
and kurtosis examination and x 2 examination. We find that stock log-return disobeys normal
distribution with one day. Then we fit to the function of the stock log-return, The first method adopt method of Mantegna and Stanley in studying the log-return of S&P500 index distribution of stock exchange of New York in 1995 to fitted to Shanghai Stock Exchange's composite index. We find stable Levy distribution can describe Shanghai Stock Exchange's composite index. The second kinds of methods are that compared Laplace distribution with normal distribution to see Laplace distribution is more superior than normal distribution. We finds that all stock log-return of day we study disobey normal distribution. More stock log-return obeye Laplace distribution. But with we enlarging interval of time, the stocks which obey normal distribution increases gradually, This is unanimous with Fama's assumption that the long-term log-return obey normal distribution. We have started put forward a simple stock price model from investor on stock market, and utilize this model to analyse the stock log-return function.
At last, we set up two kinds of models to predict the stock price, the first kind is Grey-Markov model, find it's accuracy more than Grey model, The second kind is revised arrays model of time.
本文首先讨论了股票收益分布的种类，作者对前人的工作综合发现目前描述股票收益的函数大概有七种：高斯正态分布、利维稳定分布、t标度分布、尖峰态分布、随机波动率模型、ARCH—GARCH模型、分形布朗运动，我们分别对这七种函数进行了简要的介绍。
然后对股票收益是否服从正态分布进行检验，采用偏度、峰度检验和x~2检验对上海股票市场的几支股票进行了检验，发现以天为时间尺度，股票收益并不服从正态分布。接着我们对股票收益函数进行拟合，第一种方法是采用Mantegna和Stanley在1995年研究美国纽约股票交易所S&P500指数的收益分布所采用的方法，对上证综合指数进行了拟合。发现利维稳定分布能很好的描述上证综合指数。第二种方法是对Laplace分布和正态分布进行对比，看Laplace分布是否比正态分布更优越。结果发现在我们研究的股票中，所有股票的日收益都不服从正态分布，而大多数股票收益服从Laplace分布。但是随着时间间距的放大，服从正态分布的股票逐渐增多，这与Fama关于长期收益服从正态分布的假设是一致的。我们又从股票市场上的投资者进行入手提出了一个简单的股票价格模型，并利用此模型对股票收益函数进行分析。
最后，我接着建立了两种模型对股票价格进行预测，第一种是灰色—马尔柯夫模型，发现其相对于灰色理论模型来说精确度还是比较高的；第二种是经过修正的时间序列模型。
When we study stock market, we usually think that the model of stock price obeys Brownian movement, which log-return characterize normal distribution. It is the foundation that we define a lot of financial derivative which take stock as thing marked. But Statistics result of actual market indicate most log-return of stock disobey normal distribution.
In the article we at first discuss the kind of stock log-return distribution. We find that 7 kinds function can describe stock log-return form their works: Gausses of normal distribution, Levy distribution, t distribution, spike attitude distribution, random fluctuating model, ARCH-GARCH model, divide shape Brownian movement .We have introduced brief these seven kinds of function.
Then we had examined several stocks of the stock market of Shanghai with Partial degree
and kurtosis examination and x 2 examination. We find that stock log-return disobeys normal
distribution with one day. Then we fit to the function of the stock log-return, The first method adopt method of Mantegna and Stanley in studying the log-return of S&P500 index distribution of stock exchange of New York in 1995 to fitted to Shanghai Stock Exchange's composite index. We find stable Levy distribution can describe Shanghai Stock Exchange's composite index. The second kinds of methods are that compared Laplace distribution with normal distribution to see Laplace distribution is more superior than normal distribution. We finds that all stock log-return of day we study disobey normal distribution. More stock log-return obeye Laplace distribution. But with we enlarging interval of time, the stocks which obey normal distribution increases gradually, This is unanimous with Fama's assumption that the long-term log-return obey normal distribution. We have started put forward a simple stock price model from investor on stock market, and utilize this model to analyse the stock log-return function.
At last, we set up two kinds of models to predict the stock price, the first kind is Grey-Markov model, find it's accuracy more than Grey model, The second kind is revised arrays model of time.
引文
[1] Mantegna R.N, Stanley H .E. Scaling Behavior in The Dynamics of an Economic Index. Nature, 1995
[2] Fama E.F. The Behavior of Stock Market Prices. Business, vol.38, 1965
[3] Koedijk K.G, Schafgans M.M.A and De Vries C.G. The Tail Index of Exchange Rate Returns. Internation Economics ,Vol.29, 1990
[4] Ding Z, Granger C.W.J and Engle R.F. A Long Memory Property of Stock Returns and a New Model. Empirical Finance, Vol. 1, 1993
[5] Arneodo A, Muzy J.F and Somette D. Direct Causal Cascade in The Stock Market, European Physical ,Vol.2, 1998
[6] Mantegna, Stanley. Introduction of Econophysics:Correlations and Complexity in Finance. Cambridge.UK, 1999
[7] Agen A.J. Empirical Finance, 1996,3:15
[8] Mantegna R.N. Physica A, 1991,179:232
[9] L.Bachelier. The Random Character of Stock Market Prices. MIT Press, 1964
[10] B.Mandelbrot. Fractal Market Analysis. J.Business, 1994
[11] F.Black,M.Scholes. The Mathematics of Financial Derivatives. J.Political Economy, 1973
[12] A.A.Ovehinnikov, S.F.Timashev and A.A.Belyy. Kinetics of Diffusion controlled Chemical Processes. Science, 1990
[13] Cichocki A. and Unbehauen R. Neural Networks for Optimization and Signal Processing, John Willey & Sons, Chichester, 1993
[14] Cottrell M,Girard B, Girard Y, Mangeas M and Muller C. Neural Modeling for Time Series: A Statistical Stepwise Method for Weight Elimination. IEEE Trans. Neural Networks, 1995,6(6); 1355～1363
[15] Hassoun M. Foundation of Artificial Neural Networks. MIT Press. Cambridge, 1995
[16] Kamijo K and Tanigawa T. Stock Price Pattern Recognition. A Recurrent Neural Network Approach. IEEE Int. Cone Neural Networks, 1990,(1):215～221
[17] Soros George. The Alchemy of Finance -Reading the Mind of the Market .New York: John Wiley & Sons. Inc, 1984～1985
[18] Hoptroff R G. The Principles and Practice of Time Series Forecasting and Business Modelling Using Neural Nets. Neural Computing and Applications, 1992
[19] Evangelos Simoudis. Reality Check for Data. IEEE Expert, 1996
[20] Guo Z, Uhrig R E. Using Genetic Algorithm to Select Input for Neural Networks. IEEE CS Press, 1992,252～261
[21] Adeshiyan S A. Stock Market Price Prediction Using Neural Networks. Shanghai. Shanghai University, 1994
[22] James C. Van Horne. Fundamentals of Financial Management．清华大学出版社，1999
[23] 潘正君，康立山．演化计算．清华大学出版社，1998
[24] 袁任曾．人工神经网络及其应用．清华大学出版社，1999
[25] 徐迪，马大军，李远景．神经网络在股价预测中的应用．系统工程理论与实践，1998：18(11)，111-116
[26] 费良俊．计算智能技术在金融市场分析中的应用．清华大学出版社，1996
[27] 李子奈．计量经济学方法和应用．清华大学出版社，1992
[28] 杨位钦．时间序列分析与动态数据建模．北京工业学院出版社，1986
[29] 王耀东等．经济时间序列分析．上海财经大学出版社，1996
[30] 盛骤等．概率论与数理统计．等教育出版社，1999
[31] 何晓群．现代统计分析方法与应用．中国人民大学出版社，1998
[32] 楼顺天，施阳．基于MATLAB的神经网络系统分析与设计．西安电子科技大学出版社，1999
[33] 沈世镒．神经网络系统理论及其应用．科学出版社，1998
[34] 焦李成．神经网络计算．西安电子科技大学出版社，1993
[35] 费良俊．发现金融市场预测模型的计算智能方法．软件学报，1999，4
[36] 霍学文，赵军．关于股票定价理论的发展脉络．南开经济研究，1998
[37] 蕲云江，刘林．中国股票市场CAPM的实证研究．金融研究，2001．7
[38] 赵丽红．行为金融学及其对股票价格变动的解释．证券市场导报，1998．11
[39] 陈国进，赵向红．股票定价的理论考察和实证分析．金融研究，1997．8
[40] 邓聚龙．灰色系统基本方法．华中理工大学出版社，1988
[41] 刘思锋，党耀国等．灰色系统理论及应用．科学出版社，1999
[42] 梁士楚．植物种群灰色马尔柯夫模型的初步研究．广西科学院学报，1996．3
[43] 施仁杰．马尔柯夫链基础及其应用．西安电子科技大学出版社，1992
[44] 邓聚龙．灰色控制系统．华中工学院出版社，1985
[45] 傅立．灰色系统理论及其应用．科学技术文献出版社，1992
[46] 陆明勇，杨凌，张秋文等．台湾地区强地震序列的GM模型．华南地震，2000．20
[47] 宋中民．灰色区间预测的新方法．武汉理工大学学报(交通科学与工程版)，2002．26
[2] Fama E.F. The Behavior of Stock Market Prices. Business, vol.38, 1965
[3] Koedijk K.G, Schafgans M.M.A and De Vries C.G. The Tail Index of Exchange Rate Returns. Internation Economics ,Vol.29, 1990
[4] Ding Z, Granger C.W.J and Engle R.F. A Long Memory Property of Stock Returns and a New Model. Empirical Finance, Vol. 1, 1993
[5] Arneodo A, Muzy J.F and Somette D. Direct Causal Cascade in The Stock Market, European Physical ,Vol.2, 1998
[6] Mantegna, Stanley. Introduction of Econophysics:Correlations and Complexity in Finance. Cambridge.UK, 1999
[7] Agen A.J. Empirical Finance, 1996,3:15
[8] Mantegna R.N. Physica A, 1991,179:232
[9] L.Bachelier. The Random Character of Stock Market Prices. MIT Press, 1964
[10] B.Mandelbrot. Fractal Market Analysis. J.Business, 1994
[11] F.Black,M.Scholes. The Mathematics of Financial Derivatives. J.Political Economy, 1973
[12] A.A.Ovehinnikov, S.F.Timashev and A.A.Belyy. Kinetics of Diffusion controlled Chemical Processes. Science, 1990
[13] Cichocki A. and Unbehauen R. Neural Networks for Optimization and Signal Processing, John Willey & Sons, Chichester, 1993
[14] Cottrell M,Girard B, Girard Y, Mangeas M and Muller C. Neural Modeling for Time Series: A Statistical Stepwise Method for Weight Elimination. IEEE Trans. Neural Networks, 1995,6(6); 1355～1363
[15] Hassoun M. Foundation of Artificial Neural Networks. MIT Press. Cambridge, 1995
[16] Kamijo K and Tanigawa T. Stock Price Pattern Recognition. A Recurrent Neural Network Approach. IEEE Int. Cone Neural Networks, 1990,(1):215～221
[17] Soros George. The Alchemy of Finance -Reading the Mind of the Market .New York: John Wiley & Sons. Inc, 1984～1985
[18] Hoptroff R G. The Principles and Practice of Time Series Forecasting and Business Modelling Using Neural Nets. Neural Computing and Applications, 1992
[19] Evangelos Simoudis. Reality Check for Data. IEEE Expert, 1996
[20] Guo Z, Uhrig R E. Using Genetic Algorithm to Select Input for Neural Networks. IEEE CS Press, 1992,252～261
[21] Adeshiyan S A. Stock Market Price Prediction Using Neural Networks. Shanghai. Shanghai University, 1994
[22] James C. Van Horne. Fundamentals of Financial Management．清华大学出版社，1999
[23] 潘正君，康立山．演化计算．清华大学出版社，1998
[24] 袁任曾．人工神经网络及其应用．清华大学出版社，1999
[25] 徐迪，马大军，李远景．神经网络在股价预测中的应用．系统工程理论与实践，1998：18(11)，111-116
[26] 费良俊．计算智能技术在金融市场分析中的应用．清华大学出版社，1996
[27] 李子奈．计量经济学方法和应用．清华大学出版社，1992
[28] 杨位钦．时间序列分析与动态数据建模．北京工业学院出版社，1986
[29] 王耀东等．经济时间序列分析．上海财经大学出版社，1996
[30] 盛骤等．概率论与数理统计．等教育出版社，1999
[31] 何晓群．现代统计分析方法与应用．中国人民大学出版社，1998
[32] 楼顺天，施阳．基于MATLAB的神经网络系统分析与设计．西安电子科技大学出版社，1999
[33] 沈世镒．神经网络系统理论及其应用．科学出版社，1998
[34] 焦李成．神经网络计算．西安电子科技大学出版社，1993
[35] 费良俊．发现金融市场预测模型的计算智能方法．软件学报，1999，4
[36] 霍学文，赵军．关于股票定价理论的发展脉络．南开经济研究，1998
[37] 蕲云江，刘林．中国股票市场CAPM的实证研究．金融研究，2001．7
[38] 赵丽红．行为金融学及其对股票价格变动的解释．证券市场导报，1998．11
[39] 陈国进，赵向红．股票定价的理论考察和实证分析．金融研究，1997．8
[40] 邓聚龙．灰色系统基本方法．华中理工大学出版社，1988
[41] 刘思锋，党耀国等．灰色系统理论及应用．科学出版社，1999
[42] 梁士楚．植物种群灰色马尔柯夫模型的初步研究．广西科学院学报，1996．3
[43] 施仁杰．马尔柯夫链基础及其应用．西安电子科技大学出版社，1992
[44] 邓聚龙．灰色控制系统．华中工学院出版社，1985
[45] 傅立．灰色系统理论及其应用．科学技术文献出版社，1992
[46] 陆明勇，杨凌，张秋文等．台湾地区强地震序列的GM模型．华南地震，2000．20
[47] 宋中民．灰色区间预测的新方法．武汉理工大学学报(交通科学与工程版)，2002．26