The Valuation of American Exchange Options with Ap

TheValuationofAmericanExchangeOptionswithApplicationtoRealOptionsbyPeterCarr*,CornellUniversityABSTRACTAnAmericanexchangeoptiongivesitsownertherighttoexchangeoneassetforanotheratanytimepriortoexpiration.AmodelforvaluingtheseoptionsisdevelopedusingtheGeske-JohnsonapproachforvaluingAmericanputoptions.Theformulaisshowntogeneralizemuchpreviousworkinoptionpricing.Applicationofthegeneralvaluationformulatothetimingoptionincapitalinvestmenttheoryandotherrealoptionsispresented.*AslightlylongerversionofthispaperappearedasthesecondessayinmyPh.D.dissertation\EssaysonExchangeatUCLA.Iwouldliketothankthefollowingindividualsfortheircommentsandsupport:WarrenBailey,JimBrandon,MichaelBrennan,TomCopeland,DanGalai,BobGeske,MarkGrinblatt,DavidHirshleifer,CraigHolden,EduardoSchwartz,ErikSirri,SheridanTitman,WaltTorous,BrettTrueman,andtheparticipantsoftheUCLAnanceworkshop.Theyarenotrespon-sibleforanyerrors.FinancialsupportwasprovidedbyafellowshipfromtheSocialSciencesandHumanitiesResearchCouncilofCanada,aJohnM.Olinscholarship,andanAllstateDissertationFellowship.I.INTRODUCTIONAnAmericanexchangeoptiongivesitsownertherighttoexchangeoneassetforan-otheratanytimeuptoandincludingexpiration.Margrabe(1978)valuesaEuropeanexchangeoptionwhichgivesitsownertherighttosuchanexchangeonlyatexpira-tion.MargrabealsoprovesthatexerciseofanAmericanexchangeoptionwillonlyoccuratexpirationwhenneitherunderlyingassetpaysdividends.However,whentheassettobereceivedintheexchangepayssucientlylargedividends,thereisapositiveprobabilitythatanAmericanexchangeoptionwillbeexercisedstrictlypriortoexpiration.ThispositiveprobabilityinducesadditionalvalueforanAmericanexchangeoptionoveritsEuropeancounterpart.ThepurposeofthispaperistodevelopageneralformulaforvaluingAmericanexchangeoptions.TheformulageneralizestheGeske-Johnson(1984)solutionforthevalueofanAmericanputoption.Thegeneralizationessentiallyinvolvesredeningtheexercisepricetobethepriceofatradedasset.Ifeitherassetinvolvedintheexchangehasconstantvalueovertime,thenanexchangeoptionreducestoanordinarycallorputoption.Consequently,thisgeneralformulaforAmericanexchangeoptionsmaybeusedtovaluestandardcallorputoptionsasspecialcases.Furthermore,thetimingoptioninherentinacapitalinvestmentdecisioncanalsobevalued.ThepapervaluesAmericanexchangeoptionswhenbothunderlyingassetspaydividendscontinuously.Anyassetwhosepayosaccrueovertimemaybeconsideredtoyieldacontinuouspayout(e.g.,acouponbond).Furthermore,anassetmaybehaveasifitpaysdividendsif,forexample,itfurnishesaconvenienceyieldorearnsabelowequilibriumexpectedrateofreturn.Non-tradedrealassetsmayoerabelowequilibriumreturnandmayinvolveexibilitiestoswitchoperatingmodesorexchangeoneassetforanother.Asaresult,thegeneralvaluationformulamay1beusedtovaluerealoptions.Foranalyticaltractability,thedividendsfromtheunderlyingassetsarepresumedtoprovideaconstantyield.Whenthedividendyieldontheassettobereceivedintheexchangeisstrictlypositive,Americanexchangeoptionsmaybeexercisedearly.TheGeske-JohnsonapproachisusedheretovalueanAmericanexchangeoptionbecauseitpossessestwoadvantagesoverothermethods.First,thesolutionmaybedierentiatedtoaordcomparativestaticsresults.Second,apolynomialapproxima-tiontotheexactformulaiscomputationallymoreecientthaneithernitedierencesorthebinomialmethod(seeGeskeandShastri,1985).Thepaperisorganizedasfollows.Thenextsectionreviewssomeoftherelevantoptionpricingliterature.ThevaluationformulaforanAmericanexchangeoptionisderivedinsectionIII.Thefollowingsectionthenincorporatessomepreviousresultsasspecialcasesofthegeneralsolution.ApplicationofthegeneralvaluationmodeltothetimingoptionininvestmenttheoryandotherrealoptionsisdiscussedinsectionV.Thenalsectionconcludesthepaper.II.LITERATUREREVIEWThispaperisconcernedwithvaluingexchangeoptionsondividend-payingassetswhichmayrationallybeexercisedearly.Asanintroduction,thissectionreviewspreviousworkonvaluingEuropeanexchangeoptionsandAmericanputs.Tofocusthediscussion,considertheEuropeanoptiontoexchangeassetDforassetVattimeT.AssetDisreferredtoasthedeliveryasset,andassetVtheoptionedasset.ThepayotothisEuropeanoptionatTismax(0;VTDT)whereVTandDTaretheunderlyingassets’terminalprices.SupposethattheunderlyingassetpricesVtand2DtpriortoexpirationfollowageometricBrownianmotionoftheform:dVtVt=(vv)dt+vdZvt(1)dDtDt=(dd)dt+ddZdtcovdVtVt;dDtDt!=vddt;t2[0;T];wherevanddaretheexpectedratesofreturnonthetwoassets,vanddarethecorrespondingdividendyields,2vand2daretherespectivevariancerates,anddZvtanddZdtareincrementsofstandardWienerprocessesattimet.Theratesofpricechanges,dVtVtanddDtDt,canbecorrelated,withthecovariancerategivenbyvd.Theparametersv,d,v,d,andvdareassumedtobenonnegativeconstants,althoughtheycanbeallowedtobedeterministicfunctionsoftime.Undercertainassumptions,McDonaldandSiegel(1985)showthatthevalueofaEuropeanexchangeoptiononsuchdividend-payingassetsisgivenby:e(V;D;)=VevN1(d1(Pe;2))DedN1(d2(Pe;2));(2)where:N1(d)Rd0ez2=2p2dzisthestandardunivariatenormaldistributionfunction,d1(Pe;2)ln(Pe)+2=2p2,PVDis