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1、Chapter14 TheBlack-Scholes-MertonModelOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20121TheStockPriceAssumptionConsiderastockwhosepriceisSInashortperiodoftimeoflengthDt,thereturnonthestockisnormallydistributed:wheremisexpectedreturnandsisvolatilityOptions,Futures
2、,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20122TheLognormalProperty(Equations14.2and14.3,page300)ItfollowsfromthisassumptionthatSincethelogarithmofSTisnormal,STislognormallydistributedOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20123TheLognormalDistrib
3、utionOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20124ContinuouslyCompoundedReturn(Equations14.6and14.7,page302)IfxistherealizedcontinuouslycompoundedreturnOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20125TheExpectedReturnTheexpectedvalueo
4、fthestockpriceisS0emTTheexpectedreturnonthestockism–s2/2notmThisisbecausearenotthesameOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20126mandm−s2/2Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20127mistheexpectedreturninaveryshorttime,Dt,expre
5、ssedwithacompoundingfrequencyofDtm−s2/2istheexpectedreturninalongperiodoftimeexpressedwithcontinuouscompounding(or,toagoodapproximation,withacompoundingfrequencyofDt)MutualFundReturns(SeeBusinessSnapshot14.1onpage304)Supposethatreturnsinsuccessiveyearsare15%,20%,30%,−20%and25%(ann.com
6、p.)Thearithmeticmeanofthereturnsis14%Thereturnedthatwouldactuallybeearnedoverthefiveyears(thegeometricmean)is12.4%(ann.comp.)Thearithmeticmeanof14%isanalogoustomThegeometricmeanof12.4%isanalogoustom−s2/2Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20128TheVolatil
7、ityThevolatilityisthestandarddeviationofthecontinuouslycompoundedrateofreturnin1yearThestandarddeviationofthereturninashorttimeperiodtimeDtisapproximatelyIfastockpriceis$50anditsvolatilityis25%peryearwhatisthestandarddeviationofthepricechangeinoneday?Options,Futures,andOtherDerivative
8、s,8th
