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Chapter14TheBlack-Scholes-MertonModelOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20121TheStockPriceAssumptionConsiderastockwhosepriceisSInashortperiodoftimeoflengthDt,thereturnonthestockisnormallydistributed:wheremisexpectedreturnandsisvolatilityttSSDsDmD2,Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20122TheLognormalProperty(Equations14.2and14.3,page300)ItfollowsfromthisassumptionthatSincethelogarithmofSTisnormal,STislognormallydistributedOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201232or2220220ssmssmTTSSTTSSTT,lnln,lnlnTheLognormalDistributionOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20124ESSeSSeeTTTTT()()()002221varmmsContinuouslyCompoundedReturnEquations14.6and14.7,page302)IfxistherealizedcontinuouslycompoundedreturnOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull2012521=2200ssmTxSSTxeSSTxTT,lnTheExpectedReturnTheexpectedvalueofthestockpriceisS0emTTheexpectedreturnonthestockism–s2/2notmThisisbecausearenotthesame)]/[ln()]/(ln[00SSESSETTandOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20126mandm−s2/2Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20127mistheexpectedreturninaveryshorttime,Dt,expressedwithacompoundingfrequencyofDtm−s2/2istheexpectedreturninalongperiodoftimeexpressedwithcontinuouscompounding(or,toagoodapproximation,withacompoundingfrequencyofDt)MutualFundReturns(SeeBusinessSnapshot14.1onpage304)Supposethatreturnsinsuccessiveyearsare15%,20%,30%,−20%and25%(ann.comp.)Thearithmeticmeanofthereturnsis14%Thereturnedthatwouldactuallybeearnedoverthefiveyears(thegeometricmean)is12.4%(ann.comp.)Thearithmeticmeanof14%isanalogoustomThegeometricmeanof12.4%isanalogoustom−s2/2Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20128TheVolatilityThevolatilityisthestandarddeviationofthecontinuouslycompoundedrateofreturnin1yearThestandarddeviationofthereturninashorttimeperiodtimeDtisapproximatelyIfastockpriceis$50anditsvolatilityis25%peryearwhatisthestandarddeviationofthepricechangeinoneday?Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull20129tDsEstimatingVolatilityfromHistoricalData(page304-306)1.TakeobservationsS0,S1,...,Snatintervalsoftyears(e.g.forweeklydatat=1/52)2.Calculatethecontinuouslycompoundedreturnineachintervalas:3.Calculatethestandarddeviation,s,oftheui´s4.Thehistoricalvolatilityestimateis:Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201210uSSiiiln1tssˆNatureofVolatilityVolatilityisusuallymuchgreaterwhenthemarketisopen(i.e.theassetistrading)thanwhenitisclosedForthisreasontimeisusuallymeasuredin“tradingdays”notcalendardayswhenoptionsarevaluedItisassumedthatthereare252tradingdaysinoneyearformostassetsOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201211ExampleSupposeitisApril1andanoptionlaststoApril30sothatthenumberofdaysremainingis30calendardaysor22tradingdaysThetimetomaturitywouldbeassumedtobe22/252=0.0873yearsOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201212TheConceptsUnderlyingBlack-Scholes-MertonTheoptionpriceandthestockpricedependonthesameunderlyingsourceofuncertaintyWecanformaportfolioconsistingofthestockandtheoptionwhicheliminatesthissourceofuncertaintyTheportfolioisinstantaneouslyrisklessandmustinstantaneouslyearntherisk-freerateThisleadstotheBlack-Scholes-MertondifferentialequationOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201213TheDerivationoftheBlack-ScholesDifferentialEquationOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201214.ondependencetheofridgetsThisshares:+derivative:1ofconsistingportfolioaupsetWe½2222zSƒzSSƒtSSƒtƒSSƒƒzStSSDDsDsmDDsDmDTheDerivationoftheBlack-ScholesDifferentialEquationcontinuedOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201215bygivenistimeinvalueitsinchangeThebygivenisportfolio,theofvalueTheSSƒƒtSSƒƒDDDD,TheDerivationoftheBlack-ScholesDifferentialEquationcontinuedOptions,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201216:equationaldifferentiScholes-BlackthegettoequationthisinandforsubstituteWe-Hencerate.free-riskthebemustportfoliotheonreturnThe2222r?SƒS?σSƒrStƒSƒtSSffrSSfftrDDDDDDDTheDifferentialEquationAnysecuritywhosepriceisdependentonthestockpricesatisfiesthedifferentialequationTheparticularsecuritybeingvaluedisdeterminedbytheboundaryconditionsofthedifferentialequationInaforwardcontracttheboundaryconditionisƒ=S–Kwhent=TThesolutiontotheequationisƒ=S–Ke–r(T–t)Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201217TheBlack-Scholes-MertonFormulas(Seepages313-315)Options,Futures,andOtherDerivatives,8thEdition,Copyright©JohnC.Hull201218TdTTrKSdTTrKSddNSdNeKpdNeKdNScrTrTsssss10201102210)2/2()/ln()2/2()/ln()()()()(whereTheN(x)FunctionN(x)istheprobabilitythatanormallydistri