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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