ofStockPrices(金融工程-华东师范大学汤银才).ppt

Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.1,ModeloftheBehaviorofStockPricesChapter10,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.2,CategorizationofStochasticProcesses,Discretetime;discretevariableDiscretetime;continuousvariableContinuoustime;discretevariableContinuoustime;continuousvariable,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.3,ModelingStockPrices,WecanuseanyofthefourtypesofstochasticprocessestomodelstockpricesThecontinuoustime,continuousvariableprocessprovestobethemostusefulforthepurposesofvaluingderivativesecurities,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.4,MarkovProcesses,InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegotwhereweareWewillassumethatstockpricesfollowMarkovprocesses,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.5,Weak-FormMarketEfficiency,Theassertionisthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisclearlyconsistentwithweak-formmarketefficiency,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.6,ExampleofaDiscreteTimeContinuousVariableModel,Astockpriceiscurrentlyat$40Attheendof1yearitisconsideredthatitwillhaveaprobabilitydistributionoff(40,10),wheref(m,s)isanormaldistributionwithmeanmandstandarddeviations.,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.7,Questions,Whatistheprobabilitydistributionofthechangeinstockpriceover/during2years?years?years?Dtyears?Takinglimitswehavedefinedacontinuousvariable,continuoustimeprocess,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.8,Variances&StandardDeviations,InMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthatvariancesareadditiveStandarddeviationsarenotadditive,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.9,Variances&StandardDeviations(continued),Inourexampleitiscorrecttosaythatthevarianceis100peryear.Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.(YoucansaythattheSTDis10persquarerootofyears),Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.10,AWienerProcess(Seepages220-1),WeconsideravariablezwhosevaluechangescontinuouslyThechangeinasmallintervaloftimeDtisDzThevariablefollowsaWienerprocessif1.,whereisarandomdrawingfrom(0,1).2.ThevaluesofDzforany2different(non-overlapping)periodsoftimeareindependent,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.11,PropertiesofaWienerProcess,Meanofz(T)z(0)is0Varianceofz(T)z(0)isTStandarddeviationofz(T)z(0)is,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.12,TakingLimits.,Whatdoesanexpressioninvolvingdzanddtmean?ItshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingDzandDtistrueinthelimitasDttendstozeroInthisrespect,stochasticcalculusisanalogoustoordinarycalculus,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.13,GeneralizedWienerProcesses(Seepage221-4),AWienerprocesshasadriftrate(ieaveragechangeperunittime)of0andavariancerateof1InageneralizedWienerprocessthedriftrate&thevarianceratecanbesetequaltoanychosenconstants,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.14,GeneralizedWienerProcesses(continued),ThevariablexfollowsageneralizedWienerprocesswithadriftrateofa&avariancerateofb2ifdx=adt+bdz,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.15,GeneralizedWienerProcesses(continued),MeanchangeinxintimeTisaTVarianceofchangeinxintimeTisb2TStandarddeviationofchangeinxintimeTis,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.16,TheExampleRevisited,Astockpricestartsat40&hasaprobabilitydistributionoff(40,10)attheendoftheyearIfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dzIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,10),theprocessisdS=8dt+10dz,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.17,ItoProcess(Seepages224-5),InanItoprocessthedriftrateandthevarianceratearefunctionsoftimedx=a(x,t)dt+b(x,t)dzThediscretetimeequivalentisonlytrueinthelimitasDttendstozero,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.18,WhyaGeneralizedWienerProcessisnotAppropriateforStocks,ForastockpricewecanconjecturethatitsexpectedproportionalchangeinashortperiodoftimeremainsconstantWecanalsoconjecturethatouruncertaintyastothesizeoffuturestockpricemovementsisproportionaltothelevelofthestockprice,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.19,AnItoProcessforStockPrices(Seepages225-6),wheremistheexpectedreturn,sisthevolatility.Thediscretetimeequivalentis,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.20,MonteCarloSimulation,WecansamplerandompathsforthestockpricebysamplingvaluesforeSupposem=0.14,s=0.20,andDt=0.01,then,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.21,MonteCarloSimulationOnePath(continued.SeeTable10.1),Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.22,ItosLemma(Seepages229-231),Ifweknowthestochasticprocessfollowedbyx,ItoslemmatellsusthestochasticprocessfollowedbysomefunctionG(x,t)Sinceaderivativesecurityisafunctionofthepriceoftheunderlying&time,Itoslemmaplaysanimportantpartintheanalysisofderivativesecurities,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2003,ShanghaiNormalUniversity,10.23,TaylorSeriesExpansion,ATaylorsseriesexpansionofG(x,t)gives,Options,Futures,andOtherDerivatives,4thedition2000byJohnC.HullTangYincai,2