Understanding the Binomial Option Pricing Model - Investopedia

The binomial option pricing model values options using an iterative approach utilizing multiple periods to value American options. · With the model, there are ... Options&DerivativesTrading AdvancedOptionsTradingConcepts Options&DerivativesTrading OptionsTradingStrategy&Education AdvancedOptionsTradingConcepts TableofContents Expand DeterminingStockPrices BinomialOptionsValuation Examples BinomialOptionsCalculations Black-Scholes SimpleMath This"Q"isDifferent AWorkingExample AnotherExample TheBottomLine DeterminingStockPrices Toagreeonaccuratepricingforanytradableassetischallenging—that’swhystockpricesconstantlychange.Inreality,companieshardlychangetheirvaluationsonaday-to-daybasis,buttheirstockpricesandvaluationschangenearlyeverysecond.Thisdifficultyinreachingaconsensusaboutcorrectpricingforanytradableassetleadstoshort-livedarbitrageopportunities. Butalotofsuccessfulinvestingboilsdowntoasimplequestionofpresent-dayvaluation–whatistherightcurrentpricetodayforanexpectedfuturepayoff? KeyTakeaways ThebinomialoptionpricingmodelvaluesoptionsusinganiterativeapproachutilizingmultipleperiodstovalueAmericanoptions.Withthemodel,therearetwopossibleoutcomeswitheachiteration—amoveuporamovedownthatfollowabinomialtree.Themodelisintuitiveandisusedmorefrequentlyinpracticethanthewell-knownBlack-Scholesmodel. BinomialOptionsValuation Inacompetitivemarket,toavoidarbitrageopportunities,assetswithidenticalpayoffstructuresmusthavethesameprice.Valuationofoptionshasbeenachallengingtaskandpricingvariationsleadtoarbitrageopportunities.Black-Scholesremainsoneofthemostpopularmodelsusedforpricingoptionsbuthaslimitations. Thebinomialoptionpricingmodelisanotherpopularmethodusedforpricingoptions. Examples Assumethereisacalloptiononaparticularstockwithacurrentmarketpriceof$100.Theat-the-money(ATM)optionhasastrikepriceof$100withtimetoexpiryforoneyear.Therearetwotraders,PeterandPaula,whobothagreethatthestockpricewilleitherriseto$110orfallto$90inoneyear. Theyagreeonexpectedpricelevelsinagiventimeframeofoneyearbutdisagreeontheprobabilityoftheupordownmove.Peterbelievesthattheprobabilityofthestock'spricegoingto$110is60%,whilePaulabelievesitis40%. Basedonthat,whowouldbewillingtopaymorepriceforthecalloption?PossiblyPeter,asheexpectsahighprobabilityoftheupmove. BinomialOptionsCalculations Thetwoassets,whichthevaluationdependsupon,arethecalloptionandtheunderlyingstock.Thereisanagreementamongparticipantsthattheunderlyingstockpricecanmovefromthecurrent$100toeither$110or$90inoneyearandtherearenootherpricemovespossible. Inanarbitrage-freeworld,ifyouhavetocreateaportfoliocomprisedofthesetwoassets,calloptionandunderlyingstock,suchthatregardlessofwheretheunderlyingpricegoes–$110or$90–thenetreturnontheportfolioalwaysremainsthesame.Supposeyoubuy"d"sharesofunderlyingandshortonecalloptionstocreatethisportfolio. Ifthepricegoesto$110,yourshareswillbeworth$110*d,andyou'lllose$10ontheshortcallpayoff.Thenetvalueofyourportfoliowillbe(110d-10). Ifthepricegoesdownto$90,yourshareswillbeworth$90*d,andtheoptionwillexpireworthlessly.Thenetvalueofyourportfoliowillbe(90d). Ifyouwantyourportfolio'svaluetoremainthesameregardlessofwheretheunderlyingstockpricegoes,thenyourportfoliovalueshouldremainthesameineithercase:  h ( d ) − m = l ( d ) where: h = Highest potential underlying price d = Number of underlying shares m = Money lost on short call payoff l = Lowest potential underlying price \begin{aligned}&h(d)-m=l(d)\\&\textbf{where:}\\&h=\text{Highestpotentialunderlyingprice}\\&d=\text{Numberofunderlyingshares}\\&m=\text{Moneylostonshortcallpayoff}\\&l=\text{Lowestpotentialunderlyingprice}\\\end{aligned} ​h(d)−m=l(d)where:h=Highest potential underlying priced=Number of underlying sharesm=Money lost on short call payoffl=Lowest potential underlying price​ Soifyoubuyhalfashare,assumingfractionalpurchasesarepossible,youwillmanagetocreateaportfoliosothatitsvalueremainsthesameinbothpossiblestateswithinthegiventimeframeofoneyear.  1 1 0 d − 1 0 = 9 0 d d = 1 2 \begin{aligned}&110d-10=90d\\&d=\frac{1}{2}\\\end{aligned} ​110d−10=90dd=21​​ Thisportfoliovalue,indicatedby(90d)or(110d-10)=45,isoneyeardowntheline.Tocalculateitspresentvalue,itcanbediscountedbytherisk-freerateofreturn(assuming5%).  Present Value = 9 0 d × e ( − 5 % × 1  Year ) = 4 5 × 0 . 9 5 2 3 = 4 2 . 8 5 \begin{aligned}\text{PresentValue}&=90d\timese^{(-5\%\times1\text{Year})}\\&=45\times0.9523\\&=42.85\\\end{aligned} Present Value​=90d×e(−5%×1 Year)=45×0.9523=42.85​ Sinceatpresent,theportfolioiscomprisedof½shareofunderlyingstock(withamarketpriceof$100)andoneshortcall,itshouldbeequaltothepresentvalue.  1 2 × 1 0 0 − 1 × Call Price = $ 4 2 . 8 5 Call Price = $ 7 . 1 4 , i.e. the call price of today \begin{aligned}&\frac{1}{2}\times100-1\times\text{CallPrice}=\$42.85\\&\text{CallPrice}=\$7.14\text{,i.e.thecallpriceoftoday}\\\end{aligned} ​21​×100−1×Call Price=$42.85Call Price=$7.14, i.e. the call price of today​ Sincethisisbasedontheassumptionthattheportfoliovalueremainsthesameregardlessofwhichwaytheunderlyingpricegoes,theprobabilityofanupmoveordownmovedoesnotplayanyrole.Theportfolioremainsrisk-freeregardlessoftheunderlyingpricemoves. Inbothcases(assumedtoupmoveto$110anddownmoveto$90),yourportfolioisneutraltotheriskandearnstherisk-freerateofreturn. Henceboththetraders,PeterandPaula,wouldbewillingtopaythesame$7.14forthiscalloption,despitetheirdifferingperceptionsoftheprobabilitiesofupmoves(60%and40%).Theirindividuallyperceivedprobabilitiesdon’tmatterinoptionvaluation. Supposinginsteadthattheindividualprobabilitiesmatter,arbitrageopportunitiesmayhavepresentedthemselves.Intherealworld,sucharbitrageopportunitiesexistwithminorpricedifferentialsandvanishintheshortterm. Butwhereisthemuch-hypedvolatilityinallthesecalculations,animportantandsensitivefactorthataffectsoptionspricing? Thevolatilityisalreadyincludedbythenatureoftheproblem'sdefinition.Assumingtwo(andonlytwo—hencethename“binomial”)statesofpricelevels($110and$90),volatilityisimplicitinthisassumptionandincludedautomatically(10%eitherwayinthisexample). Black-Scholes ButisthisapproachcorrectandcoherentwiththecommonlyusedBlack-Scholespricing?Optionscalculatorresults(courtesyofOIC)closelymatchwiththecomputedvalue: Unfortunately,therealworldisnotassimpleas“onlytwostates.”Thestockcanreachseveralpricelevelsbeforethetimetoexpiry. Isitpossibletoincludeallthesemultiplelevelsinabinomialpricingmodelthatisrestrictedtoonlytwolevels?Yes,itisverymuchpossible,buttounderstandittakessomesimplemathematics. SimpleMath Togeneralizethisproblemandsolution: "X"isthecurrentmarketpriceofastockand"X*u"and"X*d"arethefuturepricesforupanddownmoves"t"yearslater.Factor"u"willbegreaterthanoneasitindicatesanupmoveand"d"willliebetweenzeroandone.Fortheaboveexample,u=1.1andd=0.9. Thecalloptionpayoffsare"Pup"and"Pdn"forupanddownmovesatthetimeofexpiry. ImagebySabrinaJiang©Investopedia 2020 Ifyoubuildaportfolioof"s"sharespurchasedtodayandshortonecalloption,thenaftertime"t":  VUM = s × X × u − P up where: VUM = Value of portfolio in case of an up move \begin{aligned}&\text{VUM}=s\timesX\timesu-P_\text{up}\\&\textbf{where:}\\&\text{VUM}=\text{Valueofportfolioincaseofanupmove}\\\end{aligned} ​VUM=s×X×u−Pup​where:VUM=Value of portfolio in case of an up move​  VDM = s × X × d − P down where: VDM = Value of portfolio in case of a down move \begin{aligned}&\text{VDM}=s\timesX\timesd-P_\text{down}\\&\textbf{where:}\\&\text{VDM}=\text{Valueofportfolioincaseofadownmove}\\\end{aligned} ​VDM=s×X×d−Pdown​where:VDM=Value of portfolio in case of a down move​ Forsimilarvaluationineithercaseofpricemove:  s × X × u − P up = s × X × d − P down s\timesX\timesu-P_\text{up}=s\timesX\timesd-P_\text{down} s×X×u−Pup​=s×X×d−Pdown​  s = P up − P down X × ( u − d ) = The number of shares to purchase for = a risk-free portfolio \begin{aligned}s&=\frac{P_\text{up}-P_\text{down}}{X\times(u-d)}\\&=\text{Thenumberofsharestopurchasefor}\\&\phantom{=}\text{arisk-freeportfolio}\\\end{aligned} s​=X×(u−d)Pup​−Pdown​​=The number of shares to purchase for=a risk-free portfolio​ Thefuturevalueoftheportfolioattheendof"t"yearswillbe:  In Case of Up Move = s × X × u − P up = P up − P down u − d × u − P up \begin{aligned}\text{InCaseofUpMove}&=s\timesX\timesu-P_\text{up}\\&=\frac{P_\text{up}-P_\text{down}}{u-d}\timesu-P_\text{up}\\\end{aligned} In Case of Up Move​=s×X×u−Pup​=u−dPup​−Pdown​​×u−Pup​​  In Case of Down Move = s × X × d − P down = P up − P down u − d × d − P down \begin{aligned}\text{InCaseofDownMove}&=s\timesX\timesd-P_\text{down}\\&=\frac{P_\text{up}-P_\text{down}}{u-d}\timesd-P_\text{down}\\\end{aligned} In Case of Down Move​=s×X×d−Pdown​=u−dPup​−Pdown​​×d−Pdown​​ Thepresent-dayvaluecanbeobtainedbydiscountingitwiththerisk-freerateofreturn:  PV = e ( − r t ) × [ P up − P down u − d × u − P up ] where: PV = Present-Day Value r = Rate of return t = Time, in years \begin{aligned}&\text{PV}=e(-rt)\times\left[\frac{P_\text{up}-P_\text{down}}{u-d}\timesu-P_\text{up}\right]\\&\textbf{where:}\\&\text{PV}=\text{Present-DayValue}\\&r=\text{Rateofreturn}\\&t=\text{Time,inyears}\\\end{aligned} ​PV=e(−rt)×[u−dPup​−Pdown​​×u−Pup​]where:PV=Present-Day Valuer=Rate of returnt=Time, in years​ Thisshouldmatchtheportfolioholdingof"s"sharesatXprice,andshortcallvalue"c"(present-dayholdingof(s*X -c)shouldequatetothiscalculation.)Solvingfor"c"finallygivesitas: Note:Ifthecallpremiumisshorted,itshouldbeanadditiontotheportfolio,notasubtraction.  c = e ( − r t ) u − d × [ ( e ( − r t ) − d ) × P up + ( u − e ( − r t ) ) × P down ] c=\frac{e(-rt)}{u-d}\times[(e(-rt)-d)\timesP_\text{up}+(u-e(-rt))\timesP_\text{down}] c=u−de(−rt)​×[(e(−rt)−d)×Pup​+(u−e(−rt))×Pdown​] Anotherwaytowritetheequationisbyrearrangingit: Taking"q"as:  q = e ( − r t ) − d u − d q=\frac{e(-rt)-d}{u-d} q=u−de(−rt)−d​ Thentheequationbecomes:  c = e ( − r t ) × ( q × P up + ( 1 − q ) × P down ) c=e(-rt)\times(q\timesP_\text{up}+(1-q)\timesP_\text{down}) c=e(−rt)×(q×Pup​+(1−q)×Pdown​) Rearrangingtheequationintermsof“q”hasofferedanewperspective. Nowyoucaninterpret“q”astheprobabilityoftheupmoveoftheunderlying(as“q”isassociatedwithPupand“1-q”isassociatedwithPdn).Overall,theequationrepresentsthepresent-dayoptionprice,thediscountedvalueofitspayoffatexpiry. This"Q"isDifferent Howisthisprobability“q”differentfromtheprobabilityofanupmoveoradownmoveoftheunderlying?  VSP = q × X × u + ( 1 − q ) × X × d where: VSP = Value of Stock Price at Time  t \begin{aligned}&\text{VSP}=q\timesX\timesu+(1-q)\timesX\timesd\\&\textbf{where:}\\&\text{VSP}=\text{ValueofStockPriceatTime}t\\\end{aligned} ​VSP=q×X×u+(1−q)×X×dwhere:VSP=Value of Stock Price at Time t​ Substitutingthevalueof"q"andrearranging,thestockpriceattime"t"comesto:  Stock Price = e ( r t ) × X \begin{aligned}&\text{StockPrice}=e(rt)\timesX\\\end{aligned} ​Stock Price=e(rt)×X​ Inthisassumedworldoftwo-states,thestockpricesimplyrisesbytherisk-freerateofreturn,exactlylikearisk-freeasset,andhenceitremainsindependentofanyrisk.Investorsareindifferenttoriskunderthismodel,sothisconstitutestherisk-neutralmodel. Probability“q”and"(1-q)"areknownasrisk-neutralprobabilitiesandthevaluationmethodisknownastherisk-neutralvaluationmodel. Theexamplescenariohasoneimportantrequirement–thefuturepayoffstructureisrequiredwithprecision(level$110and$90).Inreallife,suchclarityaboutstep-basedpricelevelsisnotpossible;ratherthepricemovesrandomlyandmaysettleatmultiplelevels. Toexpandtheexamplefurther,assumethattwo-steppricelevelsarepossible.Weknowthesecondstepfinalpayoffsandweneedtovaluetheoptiontoday(attheinitialstep): ImagebySabrinaJiang©Investopedia 2020 Workingbackward,theintermediatefirststepvaluation(att=1)canbemadeusingfinalpayoffsatsteptwo(t=2),thenusingthesecalculatedfirststepvaluation(t=1),thepresent-dayvaluation(t=0)canbereachedwiththesecalculations. Togetoptionpricingatnumbertwo,payoffsatfourandfiveareused.Togetpricingfornumberthree,payoffsatfiveandsixareused.Finally,calculatedpayoffsattwoandthreeareusedtogetpricingatnumberone. Pleasenotethatthisexampleassumesthesamefactorforup(anddown)movesatbothsteps–uanddareappliedinacompoundedfashion. AWorkingExample Assumeaputoptionwithastrikepriceof$110iscurrentlytradingat$100andexpiringinoneyear.Theannualrisk-freerateis5%.Priceisexpectedtoincreaseby20%anddecreaseby15%everysixmonths. Here,u=1.2andd=0.85, x=100, t=0.5 usingtheabovederivedformulaof  q = e ( − r t ) − d u − d q=\frac{e(-rt)-d}{u-d} q=u−de(−rt)−d​ wegetq=0.35802832 valueofputoptionatpoint2,  p 2 = e ( − r t ) × ( p × P upup + ( 1 − q ) P updn ) where: p = Price of the put option \begin{aligned}&p_2=e(-rt)\times(p\timesP_\text{upup}+(1-q)P_\text{updn})\\&\textbf{where:}\\&p=\text{Priceoftheputoption}\\\end{aligned} ​p2​=e(−rt)×(p×Pupup​+(1−q)Pupdn​)where:p=Price of the put option​ AtPupup condition,underlyingwillbe=100*1.2*1.2=$144leadingtoPupup = zero AtPupdn condition,underlyingwillbe=100*1.2*0.85=$102leadingto Pupdn = $8 AtPdndn condition,underlyingwillbe=100*0.85*0.85=$72.25leadingto Pdndn = $37.75 p2=0.975309912*(0.35802832*0+(1-0.35802832)*8)=5.008970741 Similarly,p3=0.975309912*(0.35802832*8+(1-0.35802832)*37.75)=26.42958924  p 1 = e ( − r t ) × ( q × p 2 + ( 1 − q ) p 3 ) p_1=e(-rt)\times(q\timesp_2+(1-q)p_3) p1​=e(−rt)×(q×p2​+(1−q)p3​) Andhencevalueofputoption,p1=0.975309912*(0.35802832*5.008970741+(1-0.35802832)*26.42958924)=$18.29. Similarly,binomialmodelsallowyoutobreaktheentireoptiondurationtofurtherrefinedmultiplestepsandlevels.Usingcomputerprogramsorspreadsheets,youcanworkbackwardonestepatatimetogetthepresentvalueofthedesiredoption. AnotherExample AssumeaEuropean-typeputoptionwithninemonthstoexpiry,astrikepriceof$12andacurrentunderlyingpriceat$10.Assumearisk-freerateof5%forallperiods.Assumeeverythreemonths,theunderlyingpricecanmove20%upordown,givingusu=1.2,d=0.8,t=0.25andathree-stepbinomialtree. ImagebySabrinaJiang©Investopedia 2020 Redindicatesunderlyingprices,whileblueindicatesthepayoffofputoptions.  Risk-neutralprobability"q"computesto0.531446. Usingtheabovevalueof"q"andpayoffvaluesatt=ninemonths,thecorrespondingvaluesatt=sixmonthsarecomputedas: ImagebySabrinaJiang©Investopedia 2020 Further,usingthesecomputedvaluesatt=6,valuesatt=3thenatt=0are: ImagebySabrinaJiang©Investopedia 2020 Thatgivesthepresent-dayvalueofaputoptionas$2.18,prettyclosetowhatyou'dfinddoingthecomputationsusingtheBlack-Scholesmodel($2.30). TheBottomLine Althoughusingcomputerprogramscanmake theseintensivecalculationseasy,thepredictionoffuturepricesremainsamajorlimitationofbinomialmodelsforoptionpricing.Thefinerthetimeintervals,themoredifficultitgetstopredictthepayoffsattheendofeachperiodwithhigh-levelprecision. However,theflexibilitytoincorporatethechangesexpectedatdifferentperiodsisaplus,whichmakesitsuitableforpricingAmericanoptions,includingearly-exercisevaluations. ThevaluescomputedusingthebinomialmodelcloselymatchthosecomputedfromothercommonlyusedmodelslikeBlack-Scholes,whichindicates theutilityandaccuracyofbinomialmodelsforoptionpricing.Binomialpricingmodelscanbedevelopedaccordingtoatrader'spreferencesandcanworkasanalternativeto Black-Scholes. ArticleSources Investopediarequireswriterstouseprimarysourcestosupporttheirwork.Theseincludewhitepapers,governmentdata,originalreporting,andinterviewswithindustryexperts.Wealsoreferenceoriginalresearchfromotherreputablepublisherswhereappropriate.Youcanlearnmoreaboutthestandardswefollowinproducingaccurate,unbiasedcontentinour editorialpolicy. OptionsIndustryCouncil."Black-ScholesFormula."AccessedApril3,2020. CompareAccounts AdvertiserDisclosure × TheoffersthatappearinthistablearefrompartnershipsfromwhichInvestopediareceivescompensation.Thiscompensationmayimpacthowandwherelistingsappear.Investopediadoesnotincludealloffersavailableinthemarketplace. 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