Black-Scholes-Merton Model - Overview, Equation, Assumptions

Lognormal distribution: The Black-Scholes-Merton model assumes that stock prices follow a lognormal distribution based on the principle that asset prices cannot ... Black-Scholes-MertonModelApricingmodelusedtodeterminethefairpricesofstockoptionsbasedonsixvariablesHome›Resources›Knowledge›Trading&Investing›Black-Scholes-MertonModelWhatistheBlack-Scholes-MertonModel?TheBlack-Scholes-Merton(BSM)modelisapricingmodelforfinancialinstruments.Itisusedforthevaluationofstockoptions.TheBSMmodelisusedtodeterminethefairpricesofstockoptionsbasedonsixvariables:volatilityVolatilityVolatilityisameasureoftherateoffluctuationsinthepriceofasecurityovertime.Itindicatesthelevelofriskassociatedwiththepricechangesofasecurity.Investorsandtraderscalculatethevolatilityofasecuritytoassesspastvariationsintheprices,type,underlyingstockprice,strikepriceStrikePriceThestrikepriceisthepriceatwhichtheholderoftheoptioncanexercisetheoptiontobuyorsellanunderlyingsecurity,dependingon,time,andrisk-freerate.Itisbasedontheprincipleofhedgingandfocusesoneliminatingrisksassociatedwiththevolatilityofunderlyingassetsandstockoptions. TheBlack-Scholes-MertonEquationTheBlack-Scholes-Mertonmodelcanbedescribedasasecondorderpartialdifferentialequation.  Theequationdescribesthepriceofstockoptionsovertime. PricingaCallOptionThepriceofacalloptionCisgivenbythefollowingformula:  Where: PricingaPutOptionThepriceofaputoptionPisgivenbythefollowingformula:  Where:N–Cumulativedistributionfunctionofthestandardnormaldistribution. Itrepresentsastandardnormaldistributionwithmean=0andstandarddeviation=1T-t–Timetomaturity(inyears)St–SpotpriceoftheunderlyingassetK–Strikepricer–Risk-freerateÓ–Volatilityofreturnsoftheunderlyingasset AssumptionsoftheBlack-Scholes-MertonModelLognormaldistribution:TheBlack-Scholes-Mertonmodelassumesthatstockpricesfollowalognormaldistributionbasedontheprinciplethatassetpricescannottakeanegativevalue;theyareboundedbyzero.Nodividends:TheBSMmodelassumesthatthestocksdonotpayanydividendsorreturns.Expirationdate:Themodelassumesthattheoptionscanonlybeexercisedonitsexpirationormaturitydate.Hence,itdoesnotaccuratelypriceAmericanoptions.ItisextensivelyusedintheEuropeanoptionsmarket.Randomwalk:Thestockmarketisahighlyvolatileone,andhence,astateofrandomwalkRandomWalkTheoryTheRandomWalkTheoryisamathematicalmodelofthestockmarket.Thetheorypositsthatthepriceofsecuritiesmovesrandomlyisassumedasthemarketdirectioncannevertrulybepredicted.Frictionlessmarket:Notransactioncosts,includingcommissionandbrokerage,isassumedintheBSMmodel.Risk-freeinterestrate:Theinterestratesareassumedtobeconstant,hencemakingtheunderlyingassetarisk-freeone.Normaldistribution:Stockreturnsarenormallydistributed.Itimpliesthatthevolatilityofthemarketisconstantovertime.Noarbitrage:Thereisnoarbitrage.Itavoidstheopportunityofmakingarisklessprofit. LimitationsoftheBlack-Scholes-MertonModelLimitedtotheEuropeanmarket:Asmentionedearlier,theBlack-Scholes-MertonmodelisanaccuratedeterminantofEuropeanoptionprices.ItdoesnotaccuratelyvaluestockoptionsintheUS.Itisbecauseitassumesthatoptionscanonlybeexercisedonitsexpiration/maturitydate.Risk-freeinterestrates:TheBSMmodelassumesconstantinterestrates,butitishardlyeverthereality.Assumptionofafrictionlessmarket:Tradinggenerallycomeswithtransactioncostssuchasbrokeragefees,commissionCommissionCommissionreferstothecompensationpaidtoanemployeeaftercompletingatask,whichis,often,sellingacertainnumberofproductsorservices,etc.However,theBlackScholesMertonmodelassumesafrictionlessmarket,whichmeansthattherearenotransactioncosts.Itishardlyevertherealityinthetradingmarket.Noreturns:TheBSMmodelassumesthattherearenoreturnsassociatedwiththestockoptions.Therearenodividendsandnointerestearnings.However,itisnotthecaseintheactualtradingmarket.Thebuyingandsellingofoptionsareprimarilyfocusedonthereturns. MoreResourcesCFIofferstheFinancialModeling&ValuationAnalyst(FMVA)™BecomeaCertifiedFinancialModeling&ValuationAnalyst(FMVA)®CFI'sFinancialModelingandValuationAnalyst(FMVA)®certificationwillhelpyougaintheconfidenceyouneedinyourfinancecareer.Enrolltoday!certificationprogramforthoselookingtotaketheircareerstothenextlevel.Tokeeplearningandadvancingyourcareer,thefollowingresourceswillbehelpful:ContinuouslyCompoundedReturnContinuouslyCompoundedReturnContinuouslycompoundedreturniswhathappenswhentheinterestearnedonaninvestmentiscalculatedandreinvestedbackintotheaccountforaninfinitenumberofperiods.TheinterestiscalculatedontheprincipalamountandtheinterestaccumulatedoverthegivenperiodsOptions:CallsandPutsOptions:CallsandPutsAnoptionisaderivativecontractthatgivestheholdertheright,butnottheobligation,tobuyorsellanassetbyacertaindateataspecifiedprice.Risk-freeRateRisk-FreeRateTherisk-freerateofreturnistheinterestrateaninvestorcanexpecttoearnonaninvestmentthatcarrieszerorisk.Inpractice,therisk-freerateiscommonlyconsideredtoequaltotheinterestpaidona3-monthgovernmentTreasurybill,generallythesafestinvestmentaninvestorcanmake.SpotPriceSpotPriceThe spotpriceisthecurrentmarketpriceofasecurity,currency,orcommodityavailabletobebought/soldforimmediatesettlement.Inotherwords,itisthepriceatwhichthesellersandbuyersvalueanassetrightnow.CorporateFinanceTrainingAdvanceyourcareer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