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coxrossrubinstein

OptionPricing:

ASimplifiedApproach†

JohnC.Cox

MassachusettsInstituteofTechnologyandStanfordUniversity

StephenA.Ross

YaleUniversity

MarkRubinstein

UniversityofCalifornia,Berkeley

March1979（revisedJuly1979）

（publishedunderthesametitleinJournalofFinancialEconomics（September1979））

[1978winnerofthePomeranzePrizeoftheChicagoBoardOptionsExchange]

[reprintedinDynamicHedging:

AGuidetoPortfolioInsurance,editedbyDonLuskin（JohnWileyandSons1988）]

[reprintedinTheHandbookofFinancialEngineering,editedbyCliffSmithandCharlesSmithson（HarperandRow1990）]

[reprintedinReadingsinFuturesMarketspublishedbytheChicagoBoardofTrade,Vol.VI（1991）]

[reprintedinVasicekandBeyond:

ApproachestoBuildingandApplyingInterestRateModels,editedbyRiskPublications,AlanBrace（1996）]

[reprintedinTheDebtMarket,editedbyStephenRossandFrancoModigliani（EdwardLearPublishing2000）]

[reprintedinTheInternationalLibraryofCriticalWritingsinFinancialEconomics:

OptionsMarketseditedbyG.M.ConstantinidesandA..G.Malliaris（EdwardLearPublishing2000）]

Abstract

Thispaperpresentsasimplediscrete-timemodelforvaluingoptions.Thefundamentaleconomicprinciplesofoptionpricingbyarbitragemethodsareparticularlyclearinthissetting.Itsdevelopmentrequiresonlyelementarymathematics,yetitcontainsasaspeciallimitingcasethecelebratedBlack-Scholesmodel,whichhaspreviouslybeenderivedonlybymuchmoredifficultmethods.Thebasicmodelreadilylendsitselftogeneralizationinmanyways.Moreover,byitsveryconstruction,itgivesrisetoasimpleandefficientnumericalprocedureforvaluingoptionsforwhichprematureexercisemaybeoptimal.

____________________

†OurbestthanksgotoWilliamSharpe,whofirstsuggestedtoustheadvantagesofthediscrete-timeapproachtooptionpricingdevelopedhere.Wearealsogratefultoourstudentsoverthepastseveralyears.Theirfavorablereactionstothiswayofpresentingthingsencouragedustowritethisarticle.WehavereceivedsupportfromtheNationalScienceFoundationunderGrantsNos.SOC-77-18087andSOC-77-22301.

1.Introduction

Anoptionisasecuritythatgivesitsownertherighttotradeinafixednumberofsharesofaspecifiedcommonstockatafixedpriceatanytimeonorbeforeagivendate.Theactofmakingthistransactionisreferredtoasexercisingtheoption.Thefixedpriceistermedthestrikeprice,andthegivendate,theexpirationdate.Acalloptiongivestherighttobuytheshares;aputoptiongivestherighttoselltheshares.

Optionshavebeentradedforcenturies,buttheyremainedrelativelyobscurefinancialinstrumentsuntiltheintroductionofalistedoptionsexchangein1973.Sincethen,optionstradinghasenjoyedanexpansionunprecedentedinAmericansecuritiesmarkets.

Optionpricingtheoryhasalongandillustrioushistory,butitalsounderwentarevolutionarychangein1973.Atthattime,FischerBlackandMyronScholespresentedthefirstcompletelysatisfactoryequilibriumoptionpricingmodel.Inthesameyear,RobertMertonextendedtheirmodelinseveralimportantways.Thesepath-breakingarticleshaveformedthebasisformanysubsequentacademicstudies.

Asthesestudieshaveshown,optionpricingtheoryisrelevanttoalmosteveryareaoffinance.Forexample,virtuallyallcorporatesecuritiescanbeinterpretedasportfoliosofputsandcallsontheassetsofthefirm.Indeed,thetheoryappliestoaverygeneralclassofeconomicproblems—thevaluationofcontractswheretheoutcometoeachpartydependsonaquantifiableuncertainfutureevent.

Unfortunately,themathematicaltoolsemployedintheBlack-ScholesandMertonarticlesarequiteadvancedandhavetendedtoobscuretheunderlyingeconomics.However,thankstoasuggestionbyWilliamSharpe,itispossibletoderivethesameresultsusingonlyelementarymathematics.

Inthisarticlewewillpresentasimplediscrete-timeoptionpricingformula.Thefundamentaleconomicprinciplesofoptionvaluationbyarbitragemethodsareparticularlyclearinthissetting.Sections2and3illustrateanddevelopthismodelforacalloptiononastockthatpaysnodividends.Section4showsexactlyhowthemodelcanbeusedtolockinpurearbitrageprofitsifthemarketpriceofanoptiondiffersfromthevaluegivenbythemodel.Insection5,wewillshowthatourapproachincludestheBlack-Scholesmodelasaspeciallimitingcase.Bytakingthelimitsinadifferentway,wewillalsoobtaintheCox-Ross（1975）jumpprocessmodelasanotherspecialcase.

Othermoregeneraloptionpricingproblemsoftenseemimmunetoreductiontoasimpleformula.Instead,numericalproceduresmustbeemployedtovaluethesemorecomplexoptions.MichaelBrennanandEduardoSchwartz（1977）haveprovidedmanyinterestingresultsalongtheselines.However,theirtechniquesarerathercomplicatedandarenotdirectlyrelatedtotheeconomicstructureoftheproblem.Ourformulation,byitsveryconstruction,leadstoanalternativenumericalprocedurethatisbothsimpler,andformanypurposes,computationallymoreefficient.

Section6introducesthesenumericalproceduresandextendsthemodeltoincludeputsandcallsonstocksthatpaydividends.Section7concludesthepaperbyshowinghowthemodelcanbegeneralizedinotherimportantwaysanddiscussingitsessentialroleinvaluationbyarbitragemethods.

2.TheBasicIdea

SupposethecurrentpriceofastockisS=$50,andattheendofaperiodoftime,itspricemustbeeitherS*=$25orS*=$100.AcallonthestockisavailablewithastrikepriceofK=$50,expiringattheendoftheperiod.Itisalsopossibletoborrowandlendata25%rateofinterest.Theonepieceofinformationleftunfurnishedisthecurrentvalueofthecall,C.However,ifrisklessprofitablearbitrageisnotpossible,wecandeducefromthegiveninformationalonewhatthevalueofthecallmustbe!

Considerthefollowingleveredhedge:

（1）write3callsatCeach,

（2）buy2sharesat$50each,and

（3）borrow$40at25%,tobepaidbackat

theendoftheperiod.

Table1givesthereturnfromthishedgeforeachpossiblelevelofthestockpriceatexpiration.Regardlessoftheoutcome,thehedgeexactlybreaksevenontheexpirationdate.Therefore,topreventprofitablerisklessarbitrage,itscurrentcostmustbezero;thatis,

3C–100+40=0

ThecurrentvalueofthecallmustthenbeC=$20.

Table1

ArbitrageTableIllustratingtheFormationofaRisklessHedge

expirationdate

presentdate

S*=$25

S*=$100

write3calls

3C

—

–150

buy2shares

–100

50

200

borrow

40

–50

–50

total

—

—

Ifthecallwerenotpricedat$20,asureprofitwouldbepossible.Inparticular,ifC=$25,theabovehedgewouldyieldacurrentcashinflowof$15andwouldexperiencenofurthergainorlossinthefuture.Ontheotherhand,ifC=$15,thenthesamethingcouldbeaccomplishedbybuying3calls,sellingshort2shares,andlending$40.

Table1canbeinterpretedasdemonstratingthatanappropriatelyleveredpositioninstockwillreplicatethefuturereturnsofacall.Thatis,ifwebuysharesandborrowagainstthemintherightproportion,wecan,ineffect,duplicateapurepositionincalls.Inviewofthis,itshouldseemlesssurprisingthatallweneededtodeterminetheexactvalueofthecallwasitsstrikeprice,underlyingstockprice,rangeofmovementintheunderlyingstockprice,andtherateofinterest.Whatmayseemmoreincredibleiswhatwedonotneedtoknow:

amongotherthings,wedonotneedtoknowtheprobabilitythatthestockpricewillriseorfall.Bullsandbearsmustagreeonthevalueofthecall,relativetoitsunderlyingstockprice!

Thisexampleisverysimple,butitshowsseveralessentialfeaturesofoptionpricing.Andwewillsoonseethatitisnotasunrealisticasitseems.

3.TheBinomialOptionPricingFormula

Inthissection,wewilldeveloptheframeworkillustratedintheexampleintoacompletevaluationmethod.Webeginbyassumingthatthestockpricefollowsamultiplicativebinomialprocessoverdiscreteperiods.Therateofreturnonthestockovereachperiodcanhavetwopossiblevalues:

u–1withprobabilityq,ord–1withprobability1–q.Thus,ifthecurrentstockpriceisS,thestockpriceattheendoftheperiodwillbeeitheruSordS.Wecanrepresentthismovementwiththefollowingdiagram:

uSwithprobabilityq

S

dSwithprobability1–q

Wealsoassumethattheinterestrateisconstant.Individualsmayborroworlendasmuchastheywishatthisrate.Tofocusonthebasicissues,wewillcontinuetoassumethattherearenotaxes,transactioncosts,ormarginrequirements.Hence,individualsareallowedtosellshortanysecurityandreceivefulluseoftheproceeds.

Lettingrdenoteoneplustherisklessinterestrateoveroneperiod,werequireu>r>d.Iftheseinequalitiesdidnothold,therewouldbeprofitablerisklessarbitrageopportunitiesinvolvingonlythestockandrisklessborrowingandlending.

Toseehowtovalueacallonthisstock,westartwiththesimplestsituation:

theexpirationdateisjustoneperiodaway.LetCbethecurrentvalueofthecall,CubeitsvalueattheendoftheperiodifthestockpricegoestouSandCdbeitsvalueattheendoftheperiodifthestockpricegoestodS.Sincethereisnowonlyoneperiodremaininginthelifeofthecall,weknowthatthetermsofitscontractandarationalexercisepolicyimplythatCu=max[0,uS–K]andCd=max[0,dS–K].Therefore,

Cu=max[0,uS–K]withprobabilityq

C

Cd=max[0,dS–K]wi