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1、Chapter14 WienerProcessesandItô’sLemmaOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20141StochasticProcessesDescribesthewayinwhichavariablesuchasastockprice,exchangerateorinterestratechangesthroughtimeIncorporatesuncertaintiesOptions,Fut
2、ures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20142Example1Eachdayastockpriceincreasesby$1withprobability30%staysthesamewithprobability50%reducesby$1withprobability20%Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20143Example2E
3、achdayastockpricechangeisdrawnfromanormaldistributionwithmean$0.2andstandarddeviation$1Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20144Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20145MarkovProcesses(Seepages302
4、-303)InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegottowhereweareIstheprocessfollowedbythetemperatureatacertainplaceMarkov?WeassumethatstockpricesfollowMarkovprocessesOptions,Futures,andOtherDerivatives,9thEdition,Copy
5、right©JohnC.Hull20146Weak-FormMarketEfficiencyThisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisconsistentwithweak-formma
6、rketefficiencyOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20147ExampleAvariableiscurrently40ItfollowsaMarkovprocessProcessisstationary(i.e.theparametersoftheprocessdonotchangeaswemovethroughtime)Attheendof1yearthevariablewillhaveanorma
7、lprobabilitydistributionwithmean40andstandarddeviation10Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20148QuestionsWhatistheprobabilitydistributionofthestockpriceattheendof2years?½years?¼years?Dtyears?Takinglimitswehavedefinedacontinuou
8、sstochasticprocessOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20149Variances&StandardDeviationsInMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthat
