Understanding The Binomial Option Pricing Model

The Binomial Option Pricing Model is a risk-neutral method for valuing path-dependent options (e.g., American options). HomeFeaturesAboutTeamBlogGetinTouchSurveyOnlineCoursesFREEIntrotoGoogleDataStudioFREEIntroductiontoExcelBudgetModelinginExcelMenuHomeFeaturesAboutTeamBlogGetinTouchSurveyOnlineCoursesFREEIntrotoGoogleDataStudioFREEIntroductiontoExcelBudgetModelinginExcelUnderstandingtheBinomialOptionPricingModelDobromirDikovMay15,2020TheBinomialOptionPricingModelisarisk-neutralmethodforvaluingpath-dependentoptions(e.g.,Americanoptions).Itisapopulartoolforstockoptionsevaluation,andinvestorsusethemodeltoevaluatetherighttobuyorsellatspecificpricesovertime.Underthismodel,thecurrentvalueofanoptionisequaltothepresentvalueoftheprobability-weightedfuturepayoffs.ItisdifferentfromtheBlack-Scholesmodel,whichismoresuitableforpath-independentoptions,whichcannotbeexercisedbeforetheirduedate.BinomialOptionPricingModelAninvestorknowsthecurrentstockpriceatanygivenmoment.Theywilltrytoguessthestockpricemovementsinthefuture.Underthismodel,wesplitthetimetoexpirationoftheoptionintoequalperiods(weeks,months,quarters).Thenthemodelfollowsaniterativemethodtoevaluateeachperiod,consideringeitheranupordownmovementandtherespectiveprobabilities.Effectively,themodelcreatesabinomialdistributionofpossiblestockprices.It’smostlyusefulforAmerican-styleoptions,whichinvestorscanexerciseatanygiventime.Themodelalsoassumesthere’snoarbitrage,meaningthere’snobuyingwhilesellingatahigherprice.Havingno-arbitrageensuresthevalueoftheassetremainsthesame,whichisarequirementfortheBinomialOptionPricingmodeltowork.AssumptionsWhensettingupabinomialoptionpricingmodel,weneedtobeawareoftheunderlyingassumptions,tounderstandthelimitationsofthisapproachbetter.Ateverypointintime,thepricecangotoonlytwopossiblenewprices,oneupandonedown(thisisinthename,binomial);Theunderlyingassetpaysnodividends;Theinterestrate(discountfactor)isaconstantthroughouttheperiod;Themarketisfrictionless,andtherearenotransactioncostsandnotaxes;Investorsarerisk-neutral,indifferenttorisk;Therisk-freerateremainsconstant.Advantagesanddisadvantages(+)Themodelismathematicallysimpletocalculate;(+)BinomialOptionPricingisusefulforAmericanoptions,wheretheholderhastherighttoexerciseatanytimeupuntilexpiration.(-)Asignificantadvantageisamulti-periodviewthemodelprovidesfortheunderlyingasset’spriceandthetransparencyoftheoption’svalueovertime.(-)Anotabledisadvantageisthatthecomputationalcomplexityrisesalotinmulti-periodmodels.(-)Themostsignificantlimitationofthemodelistheinherentnecessitytopredictfutureprices.HowtoCalculatetheModelIfwesetthecurrent(spot)priceofanoptionasS,thenwecanhavetwopricemovementsatanygivenmoment.ThepricecaneithergouptoS+ordowntoS-.Onthisbasis,wecalculatetheup(u)anddown(d)factors.CallOptionsAcalloptionentitlesitsholdertopurchasetheunderlyingassetorstockattheexercisepricePX.Acalloptionisin-the-moneywhenthespotpriceisabovetheexerciseprice(S>PX).Whenwehaveanupmovement,thepayoffofthecalloptionisthemaximumbetweenzeroandthespotpricemultipliedbytheupfactorandreducedwiththeexerciseprice.Tovisualizethat,here’stheformula:Adownwardmovementgivesapayoffof:TheBinomialmodeleffectivelyweighsthedifferentpayoffswiththeirrespectiveprobabilitiesanddiscountsthemtothepresentvalue.PutOptionsAputoptionentitlestheholdertosellattheexercisepricePX.Whenthepricegoesdownorup,wecalculateaputoptionasfollows:BinomialTreesThebinomialtreeisthebestwaytorepresentthemodelvisually.Theyshowtheoptionpayoffandprobabilityatdifferentnodes.Nodesoutlinethepathsthepriceoftheunderlyingassetmaytakeovertime.Wecanrepresentageneralone-periodcalloptionlikethis.Wecanalsopresentitasaformula:Theputoptionusesthesameformulaasthecalloption.Where:πistheprobabilityofanupmove;risthediscountrate.Toarriveattheprobabilityofanupmoveweemploytheformula:Where:tistheperiodmultiplier(t=0.5fora6-monthperiod);risthediscountrate;disthedownfactoruistheupfactor.Inthecaseofamulti-periodoption,wecanaccumulatetheadditionalstagesbymultiplyingtheup/downfactorsforeverypricemovement.Ifwehaveanupmove,followedbyadownmove,thenwewilludSinourformulas.BinomialTreeExampleAsanexample,wecanlookatacalloptionwithsixmonthstillmaturity,andbuildabinomialtreewithaperiodofthreemonths.Wewillsetthefollowingassumptions:Tosetupourmodel,weneedtocalculatesomeparameters.Weexpectthepricetoeithergoupwith20%ordownwith10%withinasingletimestep.Applyingtheprobabilityformulafromabove,wearriveatourmodelvariables.Thenextstepistoconstructthebinomialtreeforourmodel.Wesetupthetwotime-stepsforourperiodandendupwiththreepositionsintime–present,inthreemonthsandsixmonths.Usingtheupanddownfactors,wecancalculatethestockpriceateachofthenodes.Thenextstepistocalculatetheoptionvalueattheterminaldate(t=0.50).Itequalsthemaximumofzeroandthedifferencebetweenthecurrentpriceatt=0.50andthestrikeprice.Workingbackward,wecalculatetheoptionvalueatt=0.25andthepresent.Wedothisbyweighingthepossiblefuturevalueswiththeupanddownmoveprobabilitiesanddiscountingthemwiththerisk-freerate.Thatway,wearriveatthepresentvalueoftheoption,at6.40euros.DeltaPortfolioHedgingDeltaHedgingisanotherapproachtothebinomialoptionpricingmodel.Theideaistobuildasynthetichedgeportfolioandfindtheprofitability,atwhichtheportfolioprovidesarisk-freepayoff.Thatway,wecandeterminethetradingvalueoftheportfolio,andfromthere,thepriceoftheoption.Herearetheassumptionsforourmodel:Thenextstepistocalculatetheoptionpayoffattheterminaldate.Wearelookingataone-periodbinomialmodelforthesakeofsimplicity,andouroptionvalueatperiodonewillbe.WecannowusetheoptionpayoffstocalculatetheHedgeRatio,viatheformula:Inreality,wewillmultiplythehedgeratiobysomemultipliertomakeitawholenumber,aswecannottypicallytradeinfractionsofstocks.However,forthisexample,wewillignorethatlimitationoftherealworld.Thenextstepistobuildourportfolio.Wewillbuystockandthenshorttheoption.Wecalculatetheportfoliovalueintheupanddownstatesandexpectthesetobethesame,becauseoftherisk-freepayoffassumption.Asourpayoffisrisk-free,thismeanstherateofreturnoftheportfolioistherisk-freerate.Thereforetheportfoliotodayshouldbeworththepresentvalueofthepayoff.Usingeitheroftheupordownstates,wecanapplyourrisk-freediscountfactorof10%andarriveatthecurrentvalueofthepayoffat-72.73.Ifweadjustourportfoliofunctionforthepresentweget:Ifwechangeonlyourassumptionoftheoptiontypetoacall,instead,wewillgetthesameoptionvalue,duetothefrictionlessmarketandno-arbitrageassumptions.ConclusionWecanusespreadsheetsoftwarelikeExceltomaketheBinomialOptionPricingmodelcalculationseasy,butthemajorlimitationoftheapproachremains–predictingthefutureprices.Also,aswerefinethetimestep,itgetssignificantlymoretedioustoforecasttheexpectedpayoffsattheendofeachperiod.However,theBinomialOptionPricingmodelhastheflexibilitytoaccommodatechangingcircumstancesatdifferentperiodsandthusissuitablefortheevaluationofearly-exitstrategies.ABinomialPricingmodelandaBlack-Scholesmodelgenerallyhavesimilarresults,whichisatestamenttotheviabilityoftheBinomialOptionPricingModel.Please,showyoursupportbysharingthisarticlewithcolleaguesandfriends!Also,don’tforgettodownloadtheExcelmodelbelow.Magnimetrics-Binomial-Option-Pricing-ModelFREEDOWNLOADSubscribetoourNewsletterGetaFREEExcelBenchmarkAnalysisTemplateNameEmailOpt-inIagreetoreceiveMagnimatricsnewslettersandacceptthedataprivacystatement.Imayunsubscribeatanytimeusingthelinkinthenewsletter.SubscribeDobromirDikovFCCA,FMVAHi!Iamafinanceprofessionalwith10+yearsofexperienceinaudit,controlling,reporting,financialanalysisandmodeling.Iamexcitedtodelvedeepintospecificsofvariousindustries,whereIcanidentifythebestsolutionsforclientsIworkwith.Inmysparetime,Iamintoskiing,hikingandrunning.IamalsoactiveonInstagramandYouTube,whereItrydifferentwaystoexpressmycreativeside.TwitterInstagramYoutubeLinkedinTheinformationandviewssetoutinthispublicationarethoseoftheauthor(s)anddonotnecessarilyreflecttheofficialopinionofMagnimetrics.NeitherMagnimetricsnoranypersonactingontheirbehalfmaybeheldresponsiblefortheusewhichmaybemadeoftheinformationcontainedherein.Theinformationinthisarticleisforeducationalpurposesonlyandshouldnotbetreatedasprofessionaladvice.Magnimetricsandtheauthorofthispublicationacceptnoresponsibilityforanydamagesorlossessustainedintheresultofusingtheinformationpresentedinthepublication.AnyThoughts?CancelreplyYoumightalsolikeoneofthefollowingarticles:FinancialAnalysisWhatareEmployeeStockOptions(ESOs)?Moreandmorecompaniesareofferingstockoptionsaspartofthecompensationpackagewhenrecruitingnewtalent.EmployeeStockOptionsareatypeofReadMore»DobromirDikov05/04/2021FinancialAnalysisUnderstandingtheCostofDebtRatioResearchintheSMEsectorshowsthataround40-50%ofcompaniesseekdebtfinancingatleastonceintheirlifecycle.It’sessentialtounderstandReadMore»DobromirDikov29/01/2021FinancialAnalysisUnderstandingtheGordonGrowthModelforStockValuationUnderstandingtheGordonGrowthModelforStockValuationTheGordonGrowthModel(GGM)isamethodforthevaluationofstocks.InvestorsuseittoReadMore»DobromirDikov11/12/2020HomeFeaturesAboutMagnimetricsMeettheTeamBlogGetinTouchSurveyExcelDownloadsPrivacyPolicyCookiePolicyTerms&conditionsMenuHomeFeaturesAboutMagnimetricsMeettheTeamBlogGetinTouchSurveyExcelDownloadsPrivacyPolicyCookiePolicyTerms&conditionsMagnimetricsismadewithinPlovdiv,Bulgaria.LinkedinFacebookTwitterYoutubeMediumWaitaminute!GetaFREEExcelBenchmarkAnalysisTemplatewhenyousubscribetoournewsletterNameEmailOpt-inIagreetoreceiveMagnimatricsnewslettersandacceptthedataprivacystatement.Imayunsubscribeatanytimeusingthelinkinthenewsletter.Iwantit