oftheBehaviorofStockPrices(金融工程学,华东

Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.1ModeloftheBehaviorofStockPricesChapter10Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.2CategorizationofStochasticProcessesDiscretetime;discretevariableDiscretetime;continuousvariableContinuoustime;discretevariableContinuoustime;continuousvariableOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.3ModelingStockPricesWecanuseanyofthefourtypesofstochasticprocessestomodelstockpricesThecontinuoustime,continuousvariableprocessprovestobethemostusefulforthepurposesofvaluingderivativesecuritiesOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.4MarkovProcessesInaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegotwhereweareWewillassumethatstockpricesfollowMarkovprocessesOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.5Weak-FormMarketEfficiency•Theassertionisthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.•AMarkovprocessforstockpricesisclearlyconsistentwithweak-formmarketefficiencyOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.6ExampleofaDiscreteTimeContinuousVariableModelAstockpriceiscurrentlyat$40Attheendof1yearitisconsideredthatitwillhaveaprobabilitydistributionoff(40,10),wheref(m,s)isanormaldistributionwithmeanmandstandarddeviations.Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.7Questions•Whatistheprobabilitydistributionofthechangeinstockpriceover/during2years?½years?¼years?Dtyears?Takinglimitswehavedefinedacontinuousvariable,continuoustimeprocessOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.8Variances&StandardDeviationsInMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthatvariancesareadditiveStandarddeviationsarenotadditiveOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.9Variances&StandardDeviations(continued)Inourexampleitiscorrecttosaythatthevarianceis100peryear.Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.(YoucansaythattheSTDis10persquarerootofyears)Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.10AWienerProcess(Seepages220-1)WeconsideravariablezwhosevaluechangescontinuouslyThechangeinasmallintervaloftimeDtisDzThevariablefollowsaWienerprocessif1.,whereisarandomdrawingfromf(0,1).2.ThevaluesofDzforany2different(non-overlapping)periodsoftimeareindependenttzDDOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.11PropertiesofaWienerProcessMeanof[z(T)–z(0)]is0Varianceof[z(T)–z(0)]isTStandarddeviationof[z(T)–z(0)]isTOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.12TakingLimits...Whatdoesanexpressioninvolvingdzanddtmean?ItshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingDzandDtistrueinthelimitasDttendstozeroInthisrespect,stochasticcalculusisanalogoustoordinarycalculusOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.13GeneralizedWienerProcesses(Seepage221-4)AWienerprocesshasadriftrate(ieaveragechangeperunittime)of0andavariancerateof1InageneralizedWienerprocessthedriftrate&thevarianceratecanbesetequaltoanychosenconstantsOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.14GeneralizedWienerProcesses(continued)ThevariablexfollowsageneralizedWienerprocesswithadriftrateofa&avariancerateofb2ifdx=adt+bdzOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.15GeneralizedWienerProcesses(continued)MeanchangeinxintimeTisaTVarianceofchangeinxintimeTisb2TStandarddeviationofchangeinxintimeTisDDDxatbtbTOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.16TheExampleRevisitedAstockpricestartsat40&hasaprobabilitydistributionoff(40,10)attheendoftheyearIfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dzIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,10),theprocessisdS=8dt+10dzOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity10.17ItoProcess(Seepages224-5)InanItoprocessthedriftrateandthevarianceratearefunctionsoftimedx=a(x,t)dt+b(x,t)dzThediscretetimeequivalentisonlytrueinthelimitasDttendstozeroDDDxaxttbxtt(,)(,)Option