期权、期货及其他衍生产品 课件 ull-OFOD11-ppt-14

Options,Futures,andOtherDerivativesEleventhEditionChapter14WienerProcessesandItô’sLemmaCopyright©2022,2018,2012PearsonEducation,Inc.AllRightsReservedStochasticProcessesDescribesthewayinwhichavariablesuchasastockprice,exchangerateorinterestratechangesthroughtime.Incorporatesuncertainties.Example1Eachdayastockpriceincreasesby$1withprobability30%staysthesamewithprobability50%reducesby$1withprobability20%Example2Eachdayastockpricechangeisdrawnfromanormaldistributionwithmean$0.2andstandarddeviation$1.MarkovProcessesInaMarkovprocess,futuremovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegottowhereweare.IstheprocessfollowedbythetemperatureatacertainplaceMarkov?WeassumethatstockpricesfollowMarkovprocesses.Weak-FormMarketEfficiencyThisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwords,technicalanalysisdoesnotwork.AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiency.ExampleAvariableiscurrently40.ItfollowsaMarkovprocess.Processisstationary(i.e.,theparametersoftheprocessdonotchangeaswemovethroughtime).Attheendof1year,thevariablewillhaveanormalprobabilitydistributionwithmean40andstandarddeviation10.QuestionsWhatistheprobabilitydistributionofthestockpriceattheendof2years?

Takinglimits,wehavedefinedacontinuousstochasticprocess.VariancesandStandardDeviations(1of2)InMarkovprocesses,changesinsuccessiveperiodsoftimeareindependent.Thismeansthatvariancesareadditive.Standarddeviationsarenotadditive.VariancesandStandardDeviations(2of2)Inourexample,itiscorrecttosaythatthevarianceis100peryear.Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.AWienerProcess(Equation14.1)DefineasanormaldistributionwithmeanandvarianceAvariablezfollowsaWienerprocessifThechangeinzinasmallintervaltime

Thevaluesofforany2different(non-overlapping)periodsoftimeareindependent.PropertiesofaWienerProcessMeanofVarianceofStandarddeviationofGeneralizedWienerProcesses(1of2)AWienerprocesshasadriftrate(i.e.,averagechangeperunittime)of0andavariancerateof1.InageneralizedWienerprocess,thedriftrateandthevarianceratecanbesetequaltoanychosenconstants.GeneralizedWienerProcesses(2of2)MeanchangeinxperunittimeisaVarianceofchangeinxperunittimeisTakingLimits...Whatdoesanexpressioninvolvingdzanddtmean?Itshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingandistrueinthelimitastendstozero.Inthisrespect,stochasticcalculusisanalogoustoordinarycalculus.TheExampleRevisitedAstockpricestartsat40andhasaprobabilitydistributionofattheendoftheyear.IfweassumethestochasticprocessisMarkovwithnodrift,thentheprocessisIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionistheprocesswouldbeItôProcess(Equation14.4)InanItôprocess,thedriftrateandthevarianceratearefunctionsoftimeThediscretetimeequivalentistrueinthelimitastendstozero.WhyaGeneralizedWienerProcessIsNotAppropriateforStocksForastockprice,wecanconjecturethatitsexpectedpercentage

changeinashortperiodoftimeremainsconstant(notitsexpectedactualchange).Wecanalsoconjecturethatouruncertaintyastothesizeoffuturestockpricemovementsisproportionaltothelevelofthestockprice.AnItôProcessforStockPrices

(Equations14.6and14.8)whereistheexpectedreturnisthevolatility.Thediscretetimeequivalentis:TheprocessisknownasgeometricBrownianmotion.InterestRatesWhatwouldbeareasonablestochasticprocesstoassumefortheshort-terminterestrate?MonteCarloSimulationWecansamplerandompathsforthestockpricebysamplingvaluesforSupposeandweek(=1/52or0.0192years),thenMonteCarloSimulation–SamplingOnePath(Table14.1)WeekStockPriceatStartofPeriodRandomSampleforvarepsilonChangeinStockPrice,deltaS

0100.000.522.451102.451.446.432108.88negative0.86negative3.583105.301.466.704112.00negative0.69negative2.89CorrelatedProcessesSupposeareWienerprocesseswithcorrelationThenArerandomsamplesfromabivariatestandardnormaldistributionwherecorrelationisItô’sLemma(Equation14.12)Ifweknowthestochasticprocessfollowedbyx,Itô’slemmatellsusthestochasticprocessfollowedbysomefunctionG(x,t).WhenthenSinceaderivativeisafunctionofthepriceoftheunderlyingassetandtime,Itô’slemmaplaysanimportantpartintheanalysisofderivatives.IndicationofWhyItô’sLemmaIsTrue(AppendixtoChapter14)ATaylor’sseriesexpansionofG(x,t)givesIgnoringTermsofHigherOrderThanDtInordinarycalculuswehaveInstochasticcalculusthisbecomesbecausehasacomponentwhichisoforderSubstitutingforDxSupposeSothatThenignoringtermsofhigherorderthanThee2DtTermSinceItfollowsthatThevarianceofandcanbeignored.HenceTakingLimitsTakinglimits:Substituting:Weobtain:ThisisitôsLemmaApplicationofItô’sLemmatoaStockPriceProcessThestockpriceprocessisForafunctionGofSandtExamplesTheforwardpriceofastockforacontractmaturingattimeTThelogofastockpriceFractionalBrownianMotion(1of2)InregularBrownianmotion,InfractionalBrownianmotion,whereHistheHurstexponent.FractionalBrownianMotion(2of2)WhenH>0.5,changesinsuccessiveperiodsarepositivelycorrelated.WhenH=0.5,fractionalBrownianmotionbecomesregularBrownianmotionwherechangesinsuccessiveperiodsareuncorrelated.WhenH<0.5,changesinsuccessiveperiodsarenegativelycorrelated.H=0.9(timestep=0.01)