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1、Chapter3BinomialTreeMethods------DiscreteModelsofOptionPricingAnExampleQuestion:Whent=0,buyingacalloptionofthestockatwithstrikeprice$40and1monthmaturity.Iftherisk-freeannualinterestrateis12%throughouttheperiod[0,T],howmuchshouldthepremiumforthecalloptionbe?Examplecont.1payoff=Consideraportfolio
2、Examplecont.2Whent=T,hasfixedvalue$35,nomatterSisupordownExamplecont.3Ifriskfreeinterestr=12%,abankdepositofB=35/(1+0.01)after1monthByarbitrage-freeprincipleExamplecont.4ThatisThenThisistheinvestorshouldpay$2.695forthisstockoption.AnalysisoftheExampletheideaofhedging:itispossibletoconstructanin
3、vestmentportfoliowithSandcsuchthatitisrisk-free.Theoptionpricethusdetermined(c_0=$2.695)hasnothingtodowithanyindividualinvestor'sexpectationonthefuturestockprice.One-Period&Two-StateOne-period:assetsaretradedatt=0&t=Tonly,hencethetermoneperiod.Two-state:att=TtheriskyassetShastwopossiblevalues(s
4、tates):,withtheirprobabilitiessatisfyingOne-Period&Two-StateModelThemodelisthesimplestmodel.Consideramarketconsistingoftwoassets:ariskySandarisk-freeBIf:riskyassetandriskfreeassetknown,whent=0,t=T,2possibilitiesOptionPriceatt=0?(forstrikepriceK,expiredtimeT)AnalysisoftheModel-StockPrice,isastoc
5、hasticvariableUp,withprobabilitypDown,withprobability1-pwhereisastochasticvariable.Question&AnalysisIfknownatt=T,howtofindoutwhent=0?Assumetheriskyassettobeastock.Sincethestockoptionpriceisarandomvariable,theselleroftheoptionisfacedwithariskinsellingit.However,thesellercanmanagetheriskbybuyingc
6、ertainshares(denotedasΔ)ofthestockstohedgetheriskintheoption.Thisistheidea!Δ-HedgingDefinitionDefinition:foragivenoptionV,tradeΔsharesoftheunderlyingassetSintheoppositedirection,sothattheportfolioisrisk-free.AnalysisofΔ-HedgingriskfreeassetIfΠisriskfree,then,ont=T,isriskfree.i.e.sothatAnalysiso
7、fΔ-Hedgingcont.arerandomvariables,whent=T,bothofthemhave2possiblevalueswhereareunknown,solvethem:AnalysisofΔ-Hedging (ProbabilityMeasure)DefineanewProbabilityMeasureObviouslySolutionofPremiumFromthediscussionabove,wheredenotesthee
