股票收益率的统计分析及其股价预测

广西师范大学硕士学位论文股票收益率的统计分析及其股价预测姓名：欧诗德申请学位级别：硕士专业：概率论与数理统计指导教师：杨善朝20060401股票收益率的统计分析及其股价预测2004级概率论与数理统计专业金融统计方向研究生欧诗德导师杨善朝教授摘要1973年FischerBlack和MyronScholes建立了看涨期权公式(称为Black-Scholes公式))()(2)(1dKedNSVtTrt.它基于以下的基本假设:(1)原生资产价格服从几何布朗运动tttdWdtSdS.(2)无风险利率r是常数.(3)原生资产不支付股息.(4)不支付交易费和税收.(5)不存在套利机会.自从著名的Black-Scholes公式发表以后,金融理论方面得到了飞跃发展.然而大多研究者通过对股市的研究发现股票价格并不服从几何布朗运动,即对数收益率不服从正态分布,如文献[6][7]通过对实测数据的分析,说明布朗运动与市场实际相距甚远.因此人们一直关注比较准确描绘股票价格运动的期权定价问题.由此可见对股票价格运动的研究具有重大的意义.本文并不直接研究期权定价问题,而是研究股票的对数收益率问题和股价预测问题,为投资者提供投资策略,也为今后研究期权定价做点工作.对收益率的研究必须研究收益率的分布规律和特征.由于核估计具有良好的性质：逐点渐近无偏性和一致渐近无偏性;均方相合性;强相合性.因此本文在第一章研究在不知收益率分布的情况下采用非参统计方法去估计收益率的分布和投资策略.除此之外,由于投资者还希望知道股价已涨到或跌到了某个价位时，下一步它涨到或跌到另一个价位的概率有多大.因此本文在第二章对此问题进行研究,引入周收益率和周最大收益率,用马尔可夫过程理论对上升阶段、下跌阶段与整理阶段的股票价格走势和投资策略进行全面研究分析.由于好的预测能为投资者做出好的策略,因此本文在第三章研究预测问题.ARIMA模型是基于大样本做预测分析的一个较好的模型,但多数文章只是应用这个模型,而没有处理当样本数据发生较大变化时，预测会出现大的偏差问题.本文并不用日收盘价数据对日收盘价的预测,而是用60分钟线的数据对日收盘价的预测,结果表明当日收盘价发生大波动时,预测到的日收盘价比用日收盘价数据对日收盘价的预测效果好.利用马尔可夫过程对随机现象做预测,这是一个比较好的预测方法.许多文章主要是利用一步转移概率矩阵、n步转移概率矩阵和遍历性来做预测工作,本文假设对数收益率nr服从马尔可夫过程,然后推导出股票价格nS的最优预测.GM(1,1)模型是基于小样本做预测的模型,对于具有指数变化规律的数据做预测效果显著.由于股票价格的涨跌变化可看作是能量的积存与释放过程,而能量的积存与释放一般具有指数规律,基于灰色系统理论的GM(1,1)模型恰好能反映指数的变化规律性.多数文章只是应用GM(1,1)模型解决实际问题,而本文通过用GM(1,1)模型拟合股票收盘价数据与ARIMA模型作对比,结果表明用GM(1,1)模型拟合股价数据的效果比ARIMA模型好.因此可用GM(1,1)模型做短期预测.关键词:核密度估计；几何布朗运动；转移概率矩阵;股票收益率;ARIMA模型.StatisticalAnalysisontheRateofStockReturnandPriceForecastMajor:ProbabilityTheoryandFinancialStatisticsGrade:2004Name:OUShideTutor:Prof.YANGShanchaoABSTRACTIn1973FischerBlackandMyronScholesestablishtheformulaofcalloption)()(2)(1dKedNSVtTrt,whichisknownastheformulaofBlack-Scholes.Itbasedonbelowbasicsupposition,(1)theprimarypropertypricesobeythegeometryBrownianmovementtttdWdtSdS,(2)non-riskinterestraterisconstant,(3)theprimarypropertydoesnotpaythedividend,(4)nottopaythetransactioncostandthetaxrevenue,(5)nothechanceofarbitrage.SincethefamousBlack-Scholesformulapublication,thefinancialtheoryaspectobtainedleapdevelopment.Bystudyingthestockmarket,however,manyresearchersdiscoverthatthestockpricesdonotobeythegeometryBrownianmovement,i.e.,thelogarithmreturnrateofstockpricesdonotobeynormaldistribution.Forinstance,literature[6]and[7],byanalysismeasureddata,illustratethatBrownianmovementandmarketactuallyisdistancedreallyfar.Sopeoplehavefocusedontheproblemofoptionspricingonmoreaccuratedescriptionstockpricesmovement.Thusitcanbeseenthatitismoreimportanttostudythestockpricesmovement.Thispaperdoesn’tdirectlystudytheproblemofoptionspricinginsteadofexploringtheproblemofreturnrateofstockandpriceforecasting,whichprovesinvestmentstrategyforinvestorsanddosomeworkforexploringoptionspricinginthefuture.Studyingtheruleandcharacteristicofreturnratemustbeexploredwhenwestudythereturnrate.Becausekernelestimationhasgoodnature,forexample,asymptoticunbiasedestimation,uniformasymptoticunbiasedestimation,squareconsistencyandstrongconsistency.Thusthefirstchapterofthisdocumentstudiesthedistributionofreturnratebynonparametricstatisticwayifthedistributioncan’tbefound,andprovidesinvestmentstrategyforinvestors.Withtheexceptionofthis,investorswanttoknowwhenthepricesrisetoorfalltosomepricesposition,andhowmuchtheprobabilityrisingtoorfallingtoanotherpricepositionwillbenextstep?Forstudyingtheproblem,inthesecondchapterweintroduceweekreturnrateandmaximumreturnrateinaweektoexplorethestockpricestrendandinvestmentstrategiesduringtherisestage,fallstageandtrimstagebythetheoryofMarkovprocess.Becausegoodforecastcanprovideeffectivestrategy,theproblemonforecastcanbestudiedinthethirdchapter.ARIMAmodelisabettermodelforforecastingandanalysisbasedonbigsample.Manypaperapplythemodeltodealwithrealproblem,butdon’tfocusontheproblemofdeviationwhenthesampledatahappengreatchances.Thispaperdoesn’tusethedataofdateclosingpriceinsteadofthe60-minutedatatoforecastthedateclosingprice.Theresultsindicatethattheforecastingdateclosingpricesaremoreeffectivethantheforecastingpricesbyusingthedatecloseprices.UsingMarkovprocesstoforecaststochasticphenomenonisamoreeffectivewayofforecasting.Manypapermainlyusethematrixesofone-steporn-steptransitionprobabilitiesandergodicitytoforecast.ThispapersupposereturnratenrobeyMarkovprocess,andthendeducetheoptimalforecastofstockpricesnS.GM(1,1)isakindofmodelforecastingbasedonsmallsample,whichhaseffectiveresultforthedatawithindexchangeruletoforecast.Becausetherisingorfallingchanceofstockpricescanberegardedastheprocessofenergyaccumulationandreleases,whichhastheindexchangerule,GM(1,1)basedongraysystemtheorycanexactlyreflecttheindexchangerule.AgreatdealofpaperapplytheGM(1,1)tosolveactualproblem,butthispapercomparethefitteddatabyusingGM(1,1)withARIMAmodel,theresultindicatesthatGM(1,1)ismoreeffectivethanARIMAmodel.ThusGM(1,1)canbeusedtodoforshort-termforecastingonstockmarket.Keywords:kerneldensityestimation;geometryBrownianmovement;matrixoftransitionprobabilities;stockpricereturnrate;ARIMAmodel.论文独创性声明本人郑重声明：所提交的学位论文是本人在导师的指导下进行的研究工作及取得的成果。除文中已经注明引用的内容外，本论文不含其他个人或其他机构已经发表或撰写过的研究成果。对本文的研究作出重要贡献的个人和集体，均已在文中以明确方式标明。本人承担本声明的法律责任。研究生签名：欧诗德日期：论文使用授权声明本人完全了解广西师