投资学（原书第12版）课件 英文 第21章　期权定价

ChapterTwenty-OneOptionValuationHowareoptionsvalued?ConsiderfactorsthatoughttoaffectoptionpricesExamineseveralboundswithinwhichoptionpricesmustlieQuantitativemodels“Two-state”option-valuationmodelBlack-ScholesmodelOverview©2021McGraw-HillEducation21-2IntrinsicvalueisthevaluetheoptionwouldhaveifitwereabouttoexpireSetequaltozeroforout-of-the-moneyorat-the-moneyoptionsTimevalueisthedifferencebetweenanoption’spriceandthevalueitwouldhaveifitwereexpiringimmediatelyDonotconfusetimevaluewithpresentvalueortimevalueofmoneyconceptsOptionValuation:Introduction©2021McGraw-HillEducation21-3CallOptionValueBeforeExpiration©2021McGraw-HillEducation21-4Thefollowingshouldaffectthevalueofacalloption:StockpriceExercisepriceVolatilityofstockpriceTimetoexpirationInterestrateDividendrateofstockDeterminantsofOptionValues©2021McGraw-HillEducation21-5RestrictionsPayoffiszeroatworstandhighlypositiveatbestPayoffisalwaysgreaterthanorequaltothatoftheleveragedequitypositionCallvaluecannotexceedthestockvalueRestrictionsonOptionValues:

CallOption

©2021McGraw-HillEducation21-6RangeofPossibleCallOptionValues©2021McGraw-HillEducation21-7CallOptionValueasaFunctionoftheCurrentStockPrice©2021McGraw-HillEducation21-8Calloptionholderwhowantstocloseoutthatpositionhastwochoices:ExercisethecallSellthecallCallsonnon-dividend-payingstocksare“worthmorealivethandead”Valuesofotherwise-identicalAmericanandEuropeancalloptionsonstockspayingnodividendsareequalEarlyExerciseandDividends©2021McGraw-HillEducation21-9AmericanputsareworthmorethanEuropeanputs,allelseequal,becauseAmericanputsallowyoutoexerciseanytimebeforeexpirationEarlyexercisehasvaluebecause:ThevalueofthestockcannotfallbelowzeroOncethefirmisbankrupt,itisoptimaltoexercisetheAmericanputimmediatelybecauseofthetimevalueofmoneyEarlyExerciseofAmericanPuts©2021McGraw-HillEducation21-10PutOptionValuesasaFunctionoftheCurrentStockPrice©2021McGraw-HillEducation21-11Two-StateOptionPricingAssumeastockpricecantakeonlytwopossiblevaluesatoptionexpirationThestockcouldincreasetoagivenhigherpriceThestockcoulddecreasetoagivenlowerpriceSupposethestocknowsellsatS0=$100andthepricewilleitherincreasebyafactorofu=1.20to$120orfallbyafactorofd=.9to$90byyear-endBinomialOptionPricing©2021McGraw-HillEducation21-12BinomialOptionPricing:

TextExample(1of2)©2021McGraw-HillEducation21-13BinomialOptionPricing:

TextExample(2of2)AlternativePortfolioBuy1shareofstockat$100Borrow$81.82(10%rate)Cashoutlay=$18.18PayoffFinalpayoffisexactly3timesthecalloption3C=$18.18C=$6.06©2021McGraw-HillEducation21-14HedgeratioistheratiooftherangeofthevaluesoftheoptiontothoseofthestockacrossthetwopossibleoutcomesGenerally,thehedgeratiois:HedgeRatio©2021McGraw-HillEducation21-15Binomialmodelisanoption-valuationmodelpredicatedontheassumptionthatstockpricescanmovetoonlytwovaluesoveranyshorttimeperiodByallowingforanever-greaternumberofsubperiods,wecanovercomeoneoftheapparentlimitationsofthevaluationmodel:thatthenumberofpossibleend-of-periodstockpricesissmallMakingtheValuationModelPractical(1of3)©2021McGraw-HillEducation21-16MakingtheValuationModelPractical(2of3)©2021McGraw-HillEducation21-17Beforethebinomialmodelcanbeusedtovalueactualoptions,weneedawaytochoosereasonablevaluesforuanddSpreadbetweenupanddownmovementsinpriceofthestockreflectsthevolatilityofitsrateofreturn,souanddshoulddependonthatvolatilityValuesofuanddshouldbederivedbyσ:MakingtheValuationModelPractical(3of3)©2021McGraw-HillEducation21-18Black-ScholesOptionValuation©2021McGraw-HillEducation21-19AStandardNormalCurve©2021McGraw-HillEducation21-20StockwillpaynodividendsuntilaftertheoptionexpirationdateBoththeinterestandvarianceratesofthestockareconstantInslightlymoregeneralversionsoftheformula,bothareknownfunctionsoftime,soanychangesareperfectlypredictableStockpricesarecontinuous,meaningsuddenextremejumpsareruledoutUnderlyingAssumptionsofBlack-ScholesFormula©2021McGraw-HillEducation21-21Stockvolatility,asmeasuredusingthestandarddeviationofthestockreturn,isestimatedImpliedvolatilityisthevolatilitylevelforthestockimpliedbytheoptionpriceIfactualstockstandarddeviationexceedstheimpliedvolatility,theoptionisconsideredagoodbuy.Twooptionsonthesamestock(withequalexpirationdatesbutdifferentexerciseprices)canbecompared,andtheoptionwiththehigherimpliedvolatilitywouldbeconsideredrelativelyexpensiveImpliedVolatility©2021McGraw-HillEducation21-22TheBlack-ScholescalloptionformulaassumesappliestostocksthatdonotpaydividendsbeforetheoptionexpiresWhendividendswillbepaid,weneedtoadjusttheformula(evenforEuropean-styleoptions)OneapproachistosubstituteS0e−δT

forS0intheBlack-ScholesformulaModifiedBlack-ScholesformulawouldonlyapplytoEuropeanoptionsAmericancallsondividendpayingstockswillbeworthmorethanEuropeanonesDividendsandCallOptionValuation©2021McGraw-HillEducation21-23UsingExample21.4data:C=$13.70,S0=100,r=.10,X=95,σ=.5,T=.25Wecompute:$95e-10x.25(1-.5714)-$100(1-.6664)=$6.35PutOptionValuation©2021McGraw-HillEducation21-24Hedgeratio–thesharesofstocksrequiredtohedgeagainstthepriceriskofwritingoneoptionCommonlyreferredtoastheoption’sdelta

Hedgeratioforacall:N(d1)Hedgeratioforaput:N(d1)-1Optionelasticityisthepercentageincreaseinanoption’svaluegivena1%changeinthevalueoftheunderlyingsecurityUsingtheBlack-ScholesFormula©2021McGraw-HillEducation21-25CallOptionValueandHedgeRatio©2021McGraw-HillEducation21-26Portfolioinsuranceisthepracticeofusingoptions(ordynamichedging)toprovideprotectionagainstinvestmentlosseswhilemaintainingupsidepotentialLimitationsUnlesstheinvestor’sportfoliocorrespondstoastandardmarketindexforwhichputsaretraded,aputoptionontheportfoliowillnotbeavailableIfindexputsareusedtoprotectanon-indexedportfolio,trackingerrorcanresultMaturitiesoftradedoptionsmaynotmatchtheinvestor’shorizonPortfolioInsurance©2021McGraw-HillEducation21-27Profitonanat-the-MoneyProtectivePutStrategy©2021McGraw-HillEducation21-28DeltasChangeastheStockPriceFluctuates©2021McGraw-HillEducation21-29Merton’sinsightintohowoptionpricingmodelscanprovideinsightintothefinancialcrisisof2008-2009Whenbankslend,theyimplicitlywriteaputoptiontotheborrowBorrower’sabilitytosatisfytheloanbytransferringownershipistherightto“sell”itselftothecreditorCDSprovideanevenclearerexampleWhenfirmisstrong,slopeiszero,butiffirmslipstheimplicitputrisesandslopeisnowsteeperOptionPricingandtheFinancialCrisis©2021McGraw-HillEducation21-30ValueofImplicitPutOption©2021McGraw-HillEducation21-31CallOptionElasticityasaFunctionofStockPrice©2021McGraw-HillEducation21-32OptionvalueispositivelyrelatedtovolatilityIfaninvestorbelievesimpliedvolatilityinanoption’spriceistoolow,aprofitabletradeispossibleDeltaistheslopeoftheoptionpricingcurveandtellsuspreciselyhowmanysharesofstockwemustholdtooffsetourexposureHedgingBetsOnMispricedOptions

(1of2)©2021McGraw-HillEducation21-33ProfitonaHedgedPutPortfolio©2021McGraw-HillEducation21-34ProfitisnotexactlyindependentofstockpriceAsstockpricechanges,sodothedeltasusedtocalculatethehedgeratioSensitivityofthedeltatothestockpriceiscalledthegammaoftheoptionOptiongammasareanalogoustobondconvexityHedgeratiochang