期權(quán)、期貨及其他衍生產(chǎn)品 課件 hull-OFOD11-ppt-15

Options,Futures,andOtherDerivativesEleventhEditionChapter15TheBlack–Scholes–MertonModelCopyright?2022,2018,2012PearsonEducation,Inc.AllRightsReservedTheStockPriceAssumptionConsiderastockwhosepriceisS.Inashortperiodoftimeoflengththereturnonthestockisnormallydistributed:whereisexpectedreturnandisvolatility.TheLognormalProperty(Equations15.2and15.3)ItfollowsfromthisassumptionthatSincethelogarithmofisnormal,islognormallydistributed.TheLognormalDistributionContinuouslyCompoundedReturn(Equations15.6and15.7)Ifxistherealizedcontinuouslycompoundedreturn,TheExpectedReturnTheexpectedvalueofthestockpriceisTheexpectedreturnonthestockisThisisbecausearenotthesame.muminusstartfractionsigmasquaredover2endfraction

istheexpectedreturninaveryshorttime,expressedwithacompoundingfrequencyof

istheexpectedreturninalongperiodoftimeexpressedwithcontinuouscompounding(or,toagoodapproximation,withacompoundingfrequencyofMutualFundReturns(BusinessSnapshot15.1)Supposethatreturnsinsuccessiveyearsare:Thearithmeticmeanofthereturnsis14%.Thereturnedthatwouldactuallybeearnedoverthefiveyears(thegeometricmean)is12.4%(ann.comp.)Thearithmeticmeanof14%isanalogoustoThegeometricmeanof12.4%isanalogoustoTheVolatilityThevolatilityisthestandarddeviationofthecontinuouslycompoundedrateofreturnin1year.ThestandarddeviationofthereturninashorttimeperiodtimeisapproximatelyIfastockpriceis$50anditsvolatilityis25%peryearwhatisthestandarddeviationofthepricechangeinoneday?EstimatingVolatilityFromHistoricalDataTakeobservationsatintervalsofyears(e.g.forweeklydataCalculatethecontinuouslycompoundedreturnineachintervalas:Calculatethestandarddeviation,s,oftheThehistoricalvolatilityestimateis:NatureofVolatility(BusinessSnapshot15.2)Volatilityisusuallymuchgreaterwhenthemarketisopen(i.e.theassetistrading)thanwhenitisclosed.Forthisreasontimeisusuallymeasuredin“tradingdays”notcalendardayswhenoptionsarevalued.Itisassumedthatthereare252tradingdaysinoneyearformostassets.ExampleSupposeitisApril1andanoptionlaststoApril30sothatthenumberofdaysremainingis30calendardaysor22tradingdays.ThetimetomaturitywouldbeassumedtobeTheConceptsUnderlyingBlack-Scholes-MertonTheoptionpriceandthestockpricedependonthesameunderlyingsourceofuncertainty.Wecanformaportfolioconsistingofthestockandtheoptionwhicheliminatesthissourceofuncertainty.Theportfolioisinstantaneouslyrisklessandmustinstantaneouslyearntherisk-freerate.ThisleadstotheBlack-Scholes-Mertondifferentialequation.TheDerivationoftheBlack–Scholes–MertonDifferentialEquation(Equations15.10and15.11)WesetupaportfolioconsistingofThisgetsridofthedependenceonTheDerivationoftheBlack-Scholes-MertonDifferentialEquation(Equation15.12and5.13)Thevalueoftheportfolio,isgivenbyThechangeinitsvalueintimeisgivenbyTheDerivationoftheBlack-Scholes-MertonDifferentialEquation(Equation15.15and5.16)Thereturnontheportfoliomustbetherisk-freerate.Hence,WesubstituteforinthisequationtogettheBlack-Scholesdifferentialequation:TheDifferentialEquationAnysecuritywhosepriceisdependentonthestockpricesatisfiesthedifferentialequation.Theparticularsecuritybeingvaluedisdeterminedbytheboundaryconditionsofthedifferentialequation.InaforwardcontracttheboundaryconditionisThesolutiontotheequationisPerpetualDerivative(Equation15.17)ForaperpetualderivativethereisnodependenceontimeandthedifferentialequationbecomesAderivativethatpaysoffQwhenS=Hisworth(Thesevaluessatisfythedifferentialequationandtheboundaryconditions.)TheBlack–Scholes–MertonFormulasforOptions(Equations15.20and15.21)TheNofxFunction

istheprobabilitythatanormallydistributedvariablewithameanofzeroandastandarddeviationof1islessthanx.Seetablesattheendofthebook.PropertiesofBlack–ScholesFormulaAsbecomesverylarge,

ctendstoandptendstozero.Asbecomesverysmall,c

tendstozeroandptendstoWhathappensasbecomesverylarge?WhathappensasTbecomesverylarge?UnderstandingBlack–ScholesPresentvaluefactorProbabilityofexerciseExpectedstockpriceinarisk-neutralworldifoptionisexercisedK:StrikepricepaidifoptionisexercisedRisk-NeutralValuationThevariabledoesnotappearintheBlack-Scholes-Mertondifferentialequation.Theequationisindependentofallvariablesaffectedbyriskpreference.Thesolutiontothedifferentialequationisthereforethesameinarisk-freeworldasitisintherealworld.Thisleadstotheprincipleofrisk-neutralvaluation.ApplyingRisk-NeutralValuationAssumethattheexpectedreturnfromthestockpriceistherisk-freerate.Calculatetheexpectedpayofffromtheoption.Discountattherisk-freerate.ValuingaForwardContractwithRisk-NeutralValuationPayoffisExpectedpayoffinarisk-neutralworldisPresentvalueofexpectedpayoffisProvingBlack–Scholes–MertonUsingRisk-NeutralValuation(AppendixtoChapter15)whereistheprobabilitydensityfunctionforthelognormaldistributionofinarisk-neutralworld.Wesubstitutesothatwherehistheprobabilitydensityfunctionforastandardnormal.EvaluatingtheintegralleadstotheB

S

Mresult.ImpliedVolatilityTheimpliedvolatilityofanoptionisthevolatilityforwhichtheBlack-Scholes-Mertonpriceequalsthemarketprice.Thereisaone-to-onecorrespondencebetweenpricesandimpliedvolatilities.Tradersandbrokersoftenquoteimpliedvolatilitiesratherthandollarprices.TheV

I

XS&P500VolatilityIndex(Figure15.4)Chapter26explainshowtheindexiscalculated.AnIssueofWarrantsandExecutiveStockOptionsWhenaregularcalloptionisexercisedthestockthatisdeliveredmustbepurchasedintheopenmarket.WhenawarrantorexecutivestockoptionisexercisednewTreasurystockisissuedbythecompany.Iflittleornobenefitsareforeseenbythemarket,thestockpricewillreduceatthetimetheissueisannounced.Thereisnofurtherdilution(SeeBusinessSnapshot15.3.).TheImpactofDilutionAftertheoptionshavebeenissueditisnotnecessarytotakeaccountofdilutionwhentheyarevalued.BeforetheyareissuedwecancalculatethecostofeachoptionastimesthepriceofaregularoptionwiththesametermswhereNisthenumberofexistingsharesandMisthenumberofnewsharesthatwillbecreatedifexercisetakesplace.DividendsEuropeanoptionsondividend-payingstocksarevaluedbysubstitutingthestockpricelessthepresentvalueofdividendsintoBlack-Scholes.Onlydividendswithex-dividenddatesduringlifeofoptionshouldbeincluded.The“dividend”shouldbetheexpectedreductioninthestockpriceexpected.AmericanCallsAnAmericancallonanon-dividend-payingstockshouldneverbeexercisedearly.AnAmericancallonadividend-payingstockshouldonlyeverbeexercisedimmediatelypriortoanex-dividenddate.SupposedividenddatesareattimesEarlyexerciseissometimesoptimalattimeifthedividendatthattimeisgreaterthanBlack’sApproximationforDealingWithDividendsinAmericanCallOptionsSettheAmericanpriceequalto