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1、Chapter14 WienerProcessesandItô’sLemmaOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20141StochasticProcessesDescribesthewayinwhichavariablesuchasastockprice,exchangerateorinterestratechangesthroughtimeIncorporatesuncertaintiesOptions,Futures,and
2、OtherDerivatives,9thEdition,Copyright©JohnC.Hull20142Example1Eachdayastockpriceincreasesby$1withprobability30%staysthesamewithprobability50%reducesby$1withprobability20%Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20143Example2Eachdayastockpric
3、echangeisdrawnfromanormaldistributionwithmean$0.2andstandarddeviation$1Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20144Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20145MarkovProcesses(Seepages302-303)InaMarkovprocessfut
4、uremovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegottowhereweareIstheprocessfollowedbythetemperatureatacertainplaceMarkov?WeassumethatstockpricesfollowMarkovprocessesOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20146Weak-FormMa
5、rketEfficiencyThisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiencyOptions,Futures,andOtherDe
6、rivatives,9thEdition,Copyright©JohnC.Hull20147ExampleAvariableiscurrently40ItfollowsaMarkovprocessProcessisstationary(i.e.theparametersoftheprocessdonotchangeaswemovethroughtime)Attheendof1yearthevariablewillhaveanormalprobabilitydistributionwithmean40andstandarddev
7、iation10Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20148QuestionsWhatistheprobabilitydistributionofthestockpriceattheendof2years?½years?¼years?Dtyears?TakinglimitswehavedefinedacontinuousstochasticprocessOptions,Futures,andOtherDerivatives,9t
8、hEdition,Copyright©JohnC.Hull20149Variances&StandardDeviationsInMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthat
