Hull经典衍生品教科书第9版课件

Chapter14WienerProcessesandItsLemma,StochasticProcesses,Describesthewayinwhichavariablesuchasastockprice,exchangerateorinterestratechangesthroughtimeIncorporatesuncertainties,Example1,Eachdayastockpriceincreasesby$1withprobability30%staysthesamewithprobability50%reducesby$1withprobability20%,Example2,Eachdayastockpricechangeisdrawnfromanormaldistributionwithmean$0.2andstandarddeviation$1,MarkovProcesses(Seepages302-303),InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegottowhereweareIstheprocessfollowedbythetemperatureatacertainplaceMarkov?WeassumethatstockpricesfollowMarkovprocesses,Weak-FormMarketEfficiency,Thisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiency,Example,Avariableiscurrently40ItfollowsaMarkovprocessProcessisstationary(i.e.theparametersoftheprocessdonotchangeaswemovethroughtime)Attheendof1yearthevariablewillhaveanormalprobabilitydistributionwithmean40andstandarddeviation10,Questions,Whatistheprobabilitydistributionofthestockpriceattheendof2years?years?years?Dtyears?Takinglimitswehavedefinedacontinuousstochasticprocess,Variances&StandardDeviations,InMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthatvariancesareadditiveStandarddeviationsarenotadditive,Variances&StandardDeviations(continued),Inourexampleitiscorrecttosaythatthevarianceis100peryear.Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.,AWienerProcess(Seepages304-305),Definef(m,v)asanormaldistributionwithmeanmandvariancevAvariablezfollowsaWienerprocessifThechangeinzinasmallintervaloftimeDtisDzThevaluesofDzforany2different(non-overlapping)periodsoftimeareindependent,PropertiesofaWienerProcess,Meanofz(T)z(0)is0Varianceofz(T)z(0)isTStandarddeviationofz(T)z(0)is,GeneralizedWienerProcesses(Seepage305-308),AWienerprocesshasadriftrate(i.e.averagechangeperunittime)of0andavariancerateof1InageneralizedWienerprocessthedriftrateandthevarianceratecanbesetequaltoanychosenconstants,GeneralizedWienerProcesses(continued),MeanchangeinxperunittimeisaVarianceofchangeinxperunittimeisb2,TakingLimits.,Whatdoesanexpressioninvolvingdzanddtmean?ItshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingDzandDtistrueinthelimitasDttendstozeroInthisrespect,stochasticcalculusisanalogoustoordinarycalculus,TheExampleRevisited,Astockpricestartsat40andhasaprobabilitydistributionoff(40,100)attheendoftheyearIfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dzIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,100),theprocesswouldbedS=8dt+10dz,ItProcess(Seepages308),InanItprocessthedriftrateandthevarianceratearefunctionsoftimedx=a(x,t)dt+b(x,t)dzThediscretetimeequivalentistrueinthelimitasDttendstozero,WhyaGeneralizedWienerProcessIsNotAppropriateforStocks,Forastockpricewecanconjecturethatitsexpectedpercentagechangeinashortperiodoftimeremainsconstant(notitsexpectedactualchange)Wecanalsoconjecturethatouruncertaintyastothesizeoffuturestockpricemovementsisproportionaltothelevelofthestockprice,AnItoProcessforStockPrices(Seepages308-311),wheremistheexpectedreturnsisthevolatility.ThediscretetimeequivalentisTheprocessisknownasgeometricBrownianmotion,InterestRates,Whatwouldbeareasonablestochasticprocesstoassumefortheshort-terminterestrate?,MonteCarloSimulation,WecansamplerandompathsforthestockpricebysamplingvaluesforeSupposem=0.15,s=0.30,andDt=1week(=1/52or0.192years),then,MonteCarloSimulationSamplingonePath(SeeTable14.1,page311),CorrelatedProcesses,Supposedz1anddz2areWienerprocesseswithcorrelationrThen,ItsLemma(Seepages313-315),Ifweknowthestochasticprocessfollowedbyx,ItslemmatellsusthestochasticprocessfollowedbysomefunctionG(x,t).Whendx=a(x,t)dt+b(x,t)dzthenSinceaderivativeisafunctionofthepriceoftheunderlyingassetandtime,Itslemmaplaysanimportantpartintheanalysi