The Black-Scholes formula, explained | by Jørgen Veisdal

The Black–Scholes model is a mathematical model simulating the dynamics of a financial market containing derivative financial instruments. SigninLatestMostPopularAllstoriesAboutWriteforCantor'sParadisePrivatdozentNewsletterTheBlack-Scholesformula,explainedIntroductiontothemostfamousequationinfinanceJørgenVeisdalFollowJul14,2019·12minreadTheBlack–Scholesmodelisamathematicalmodelsimulatingthedynamicsofafinancialmarketcontainingderivativefinancialinstruments.Sinceitsintroductionin1973andrefinementinthe1970sand80s,themodelhasbecomethede-factostandardforestimatingthepriceofstockoptions.Thekeyideabehindthemodelistohedgetheoptionsinaninvestmentportfoliobybuyingandsellingtheunderlyingasset(suchasastock)injusttherightwayandasaconsequence,eliminaterisk.Themethodhaslaterbecomeknownwithinfinanceas“continuouslyreviseddeltahedging”,andbeenadoptedbymanyoftheworld’sforemostinvestmentbanksandhedgefunds.ThegoalofthisarticleistoexplaintheBlack-Scholesequation’smathematicalfoundation,underlyingassumptionsandimplications.Happyreading!TheBlack-ScholesmodelTheBlack–Scholesmodelisamathematicalmodelsimulatingthedynamicsofafinancialmarketcontainingderivativefinancialinstrumentssuchasoptions,futures,forwardsandswaps.Thekeypropertyofthemodelisthatitshowsthatanoptionhasauniquepriceregardlessoftheriskoftheunderlyingsecurityanditsexpectedreturn.Themodelisbasedonapartialdifferentialequation(PDE),theso-calledBlack-Scholesequation,fromwhichonecandeducetheBlack-Scholesformula,whichgivesatheoreticalestimateofthecorrectpriceofEuropeanstockoptions.AssumptionsTheoriginalBlack-Scholesmodelisbasedonacoreassumptionthatthemarketconsistsofatleastoneriskyasset(suchasastock)andone(essentially)risk-freeasset,suchasamoneymarketfund,cashoragovernmentbond.Inaddition,itassumesthreepropertiesofthetwoassets,andfourofthemarketitself:Assumptionsabouttheassetsinthemarketare:1.Therateofreturnontherisk-freeassetisconstant(thuseffectivelybehavesasaninterestrate);2.Theinstantaneouslogreturnoftheriskyasset’spriceisassumedtobehaveasaninfinitesimalrandomwalkwithconstantdriftandvolatility,moreprecisely,accordingtogeometricBrownianmotion.3.Theriskyassetdoesnotpayadividend.Assumptionsaboutthemarketitselfare:1.Therearenoarbitrage(risk-freeprofit)opportunities;2.Itispossibletoborrowandlendanyamountofcashatthesamerateastheinterestrateoftherisk-freeasset;3.Itispossibletobuyandsellanyamountofthestock(includingshortselling);and4.Therearenotransactioncostsinthemarket(i.e.nocommissionforbuyingorsellingsecuritiesorderivativeinstruments).Insubsequentextensionsoftheoriginalmodel,theseassumptionshavebeenrevisedtoadjustfordynamicinterestratesfortherisk-freeasset(Merton,1976),transactioncostsforbuyingandselling(Ingersoll,1976)anddividendpayoutsfortheriskyasset(Whaley,1981).Inthisessay,assumeweareworkingwiththeoriginalmodel,unlessstatedotherwise.TheBlack-ScholesequationFigure1.VisualrepresentationofEuropeancalloptionprice/valuewithrespecttostrikepriceandstockprice,ascalculatedusingtheBlack-ScholesequationTheBlack-Scholesequationisthepartialdifferentialequation(PDE)thatgovernsthepriceevolutionofEuropeanstockoptionsinfinancialmarketsoperatingaccordingtothedynamicsoftheBlack-Scholes(sometimesBlack-Scholes-Merton)model.Theequationis:Equation1.TheBlack-ScholespartialdifferentialequationdescribingthepriceofaEuropeancallorputoptionovertimeWhereVisthepriceoftheoption(asafunctionoftwovariables:thestockpriceSandtimet),ristherisk-freeinterestrate(thinkinterestrateakintothatwhichyouwouldreceivefromamoney-marketfund,Germangovernmentdebtorsimilar“safe”debtsecurities)andσisthevolatilityofthelogreturnsoftheunderlyingsecurity(forthepurposesofthisarticle,weareconsideringstocks).AneatderivationoftheequationisavailableonWikipedia,basedonJohnC.Hull’s“Option,FuturesandOtherDerivatives”(1989).IfwerewritetheequationtothefollowingformEquation2.RewrittenformoftheBlack-ScholesequationThentheleftsiderepresentsthechangeinthevalue/priceoftheoptionVduetotimetincreasing+theconvexityoftheoption’svaluerelativetothepriceofthestock.Therighthandsiderepresentstherisk-freereturnfromalongpositionintheoptionandashortpositionconsistingof∂V/∂Ssharesofthestock.Intermsofthegreeks:Equation3.Theta(Θ)+Gamma(Γ)=(risk-freerate)x(priceoftheoption)-(risk-freerate)x(priceofstock)xDelta(Δ)ThekeyobservationofBlackandScholes(1973)wasthattherisk-freereturnofthecombinedportfolioofstocksandoptionsontherighthandsideoveranyinfinitesimaltimeintervalcouldbeexpressedasthesumoftheta(Θ)andatermincorporatinggamma(Γ).Theobservationissometimesknownasthe“riskneutralargument”.Thisbecausethevalueoftheta(Θ)istypicallynegative(becausethevalueoftheoptiondecreasesastimemovesclosertoexpiration)andthevalueofgamma(Γ)istypicallypositive(reflectingthegainstheportfolioreceivesfromholdingtheoption).Insum,thelossesfromthetaandthegainsfromgammaoffsetoneanother,resultinginreturnsatarisk-freerate.TheBlack-ScholesformulaTheBlack-ScholesformulaisasolutiontotheBlack-ScholesPDE,giventheboundaryconditionsbelow(eq.4and5).ItcalculatesthepriceofEuropeanputandcalloptions.Thatis,itcalculatesthepriceofcontractsfortheright(butnotobligation)tobuyorsellsomeunderlayingassetatapre-determinedpriceonapre-determineddateinthefuture.Atmaturity/expiration(T),thevalueofsuchEuropeancall(C)andput(P)optionsaregivenby,respectively:Equation4forthevalue/priceofaEuropeancalloptionEquation5forthevalue/priceofaEuropeanputoptionBlackandScholesshowedthatthefunctionalformoftheanalyticsolutiontotheBlack-Scholesequation(eq.1above)withtheboundaryconditionsgivenbyeq.4and5,foraEuropeancalloptionis:Equation6.TheBlack-ScholesformulaforthevalueofacalloptionCforanon-dividendpayingstockofpriceSTheformulagivesthevalue/priceofEuropeancalloptionsforanon-dividend-payingstock.ThefactorsgoingintotheformulaareS=priceofsecurity,T=dateofexpiration,t=currentdate,X=exerciseprice,r=risk-freeinterestrateandσ=volatility(standarddeviationoftheunderlyingasset).ThefunctionN(・)representsthecumulativedistributionfunctionforanormal(Gaussian)distributionandmaybethoughtofas‘theprobabilitythatarandomvariableislessorequaltoitsinput(i.e.d₁andd₂)foranormaldistribution’.Beingaprobability,theofvalueN(・)inotherwordswillalwaysbebetween0≤N(・)≤1.Theinputsd₁andd₂aregivenby:Equation7Veryinformally,thetwotermsinthesumgivenbytheBlack-Scholesformulamaybethoughtofas‘thecurrentpriceofthestockweightedbytheprobabilitythatyouwillexerciseyouroptiontobuythestock’minus‘thediscountedpriceofexercisingtheoptionweightedbytheprobabilitythatyouwillexercisetheoption’,orsimply‘whatyouaregoingtoget’minus‘whatyouaregoingtopay’(Khan,2013).ForaEuropeanputoption(contractsfortheright,butnotobligation,tosellsomeunderlayingassetatapre-determinedpriceonapre-determineddateinthefuture)theequivalentfunctionalformis:Equation9.TheBlack-ScholesformulaforthevalueofaputoptionCforanon-dividendpayingstockofpriceSExample:CalculatingthepriceofaEuropeancalloptionInordertocalculatewhatthepriceofaEuropeancalloptionshouldbe,weknowweneedfivevaluesrequiredbyequation6above.Theyare:1.Thecurrentpriceofthestock(S),2.Theexercisepriceofthecalloption(X),3.Thetimetoexpiration(T-t),4.Therisk-freeinterestrate(r)and5.Thevolatilityofthestock,givenbythestandarddeviationofhistoricallogreturns(σ).EstimatingthevalueofacalloptionforTesla(TSLA)Thefirstfourvaluesweneedareeasilyobtainable.Let’ssayweareinterestedinacalloptionforTesla’sstock($TSLA),maturingthedayofitsQ3earningsin2019,ataprice20%higherthanthestockiscurrentlytrading.LookingatTesla’sNASDAQlisting($TSLA)onYahooFinancetoday(July13th,2019),wefindastockpriceofS=$245.Multiplyingthecurrentpricewith1.2givesusanexerciseprice20%higherthanthestockiscurrentlytrading,X=$294.Googling,wefindthatthedayofitsQ3earningscallisOctober22nd,givingusatimetoexpiration/maturityofOct22nd-July13th=101days.Asaproxyforarisk-freeinterestrateinstrument,we’lluseUS10-yeargovernmentbonds($USGG10YR),currentlypayingoff2.12%.So,wefindS=245,X=294,T-t=101andr=0.0212.Theonlymissingvalueisanestimationofthestock’svolatility(σ).Wecanestimateanystock’svolatilitybyobservingitshistoricalprices,or,evensimpler,bycalculatingotheroptionpricesforthesamestockatdifferentmaturity/expirationdates(T)andexercise/strikeprices(X),ifweknowtheyhavebeensetaccordingtoaBlack-Scholesmodel.Theresultingvalue,σ,isanumberbetween0and1,representingthemarket’simpliedvolatilityforthestock.ForTesla,atthetimeofwritingthisarticle,thevalueaveragedatapproximately0.38for4–5differentoptionpricesaroundthesameexpiry/maturitydate.Inputintoequation6above,wefindthatthecalloptionwe’reinterestedinshouldbepricedsomewherearound$7.ImpliedvolatilityAlthoughitisinterestingtounderstandhowoptionsissuersarriveatthepriceoftheircallandputoptions,asinvestorsit’shardto“disagree”withsuchprices,perse,andsodifficulttoturnthisknowledgeintoactionableinvestmenttheses.WecanhowevergetalotofmilageoutoftheBlack-Scholesformulaifweinsteadtreatthepriceofanoption(CorP)asaknownquantity/independentvariable(foundbylookingatdifferentmaturity/expirationdatesTanddifferentexercisepricesX).Thisbecause,ifwedo,theBlack-Scholesfunctionalequationbecomesatooltohelpusunderstandhowthemarketestimatesthevolatilityofastock,alsoknownastheimpliedvolatilityoftheoption.Thisisinformationwecandisagreeover,andtradeagainst.HypotheticalscenarioIfweforinstancelookatthechartfortheTeslastockoverthelastthreemonths(figure2),weseearather(foralackofabetterword)volatilejourneyfromhoveringaround$280threemonthsago,toalowof$180amonthandahalfago,tonowonitswaybackupat$245.Thismakessensegiventhevolatilityweobservedfromcallpricesbefore($280–$180=$100,$100/280=0.36,vs0.38).Itdoesnotmakesense,however,ifwethinkthefluctuationoverthepastthreemonthswasthemeretipofaniceberg,goingintoaperiodofmorevolatilityforTesla,say,duetoanupcomingincreaseinshort-selling.Figure2.3monthchartfor$TSLALet'ssaywedisagreewithanoptionsissuerabouttheimpliedvolatilityofstock'sperformanceoverthelastthreemonths.Wethinktherideisgoingtogetrockier.Howmuch?Let'ssaythatinsteadof40%,wethinkthenextthreemonthswilllookmorelike60%.InputintothefunctionalBlack-ScholesformulaalongwiththesamevaluesforS,X,r,andT-t,wegetapriceofnearlytwiceofwhattheoptionsissuerwants,atC(S,t)=$14.32.Thiswecantradeon.Wecould,forinstance,buycalloptionstodayandwaitforvolatilitytoincreaseorthevalueofthestocktogoup,beforesellingataprofit.AmericanoptionsBecauseAmericanoptionscanbeexercisedatanydatepriortoexpiration(so-called“continuoustimelineinstruments”),theyaremuchmoredifficulttodealwiththatEuropeanoptions(“pointintimeinstruments”).Primarily,sincetheoptimalexercisepolicywillaffectthevalueoftheoption,thisneedstobetakenintoaccountwhensolvingtheBlack-Scholespartialdifferentialequation.Therearenoknown“closedform”solutionsforAmericanoptionsaccordingtotheBlack-Scholesequation.Thereare,though,somespecialcases:ForAmericancalloptionsonunderlyingassetsthatdonotpaydividend(orotherpayouts),theAmericancalloptionpriceisthesameasforEuropeancalloptions.Thisbecausetheoptimalexercisepolicyinthiscaseistonotexercisetheoption.ForAmericancalloptionsonunderlyingassetsthatdopayoneknowndividendinitslifetime,itmaybeoptimaltoexercisetheoptionearly.Insuchcasestheoptionmaybeoptimallyexercisedjustbeforethestockgoesex-dividend,accordingtoasolutiongiveninclosed-formbytheso-calledRoll-Geske-Whaleymethod(Roll,1977;Geske,1979;1981;Whaley,1981):First,checkifitisoptimaltoexercisetheoptionearly,byinvestigatingwhetherthefollowinginequalityisfulfilled:Equation10.ForS=stockprice,X=exerciseprice,D₁=dividendpaid,t=currentdate,t₁=dateofdividendpayment,T=expirationdateofoption.Iftheinequalityisnotfulfilled,earlyexerciseitnotoptimal.IfC(・)istheregularBlack-ScholesformulaforEuropeancalloptionsonnon-dividend-payingstock(eqx),thevalueoftheAmericancalloptionisthengivenbyaversionofthesameequationwherethestockprice(S)isdiscounted:Equation11.ThevalueofanAmericancalloptionwheninequality(eq.8)isnotfulfilledIftheinequalityisfulfilled,earlyexerciseisoptimalandthevalueoftheAmericancalloptionisgivenbythefollowing,awful,messofanequation(Itriedtobreakitupbyeachtermtomakeitmorereadable):Equation12.ThevalueofanAmericancalloptionwheninequality(eq.10)isfulfilledWhereasbeforeS=priceofstock,T=dateofexpirationofoption,X=exercisepriceandr=risk-freeinterestrate,σ=volatility(standarddeviationofthelogofthehistoricalreturnsofthestock),andD₁isthedividendpayout.Inaddition,ρisgivenby:Equation13.a₁,a₂by:Equation14.Equation15.andb₁,b₂by:Equation16.Equation17.LimitationsItshouldgowithoutsayingthatBlack-Scholesmodelispreciselythat,atheoreticalmodelthattriestoestimatehowamarketbehaves,giventheassumptionsstatedaboveandtheinherentlimitationsofourownnumericalestimationsofrisk-freeinterestrates(r)andfuturevolatility(σ).Itshouldherebehighlightedthatnotalltheassumptionsof(especiallytheoriginalmodel)areinfactempiricallyvalid.Forinstance,significantlimitationsarisefrom:Theunderestimationofextrememovesinthestock,yieldingtailriskTheassumptionofinstant,cost-lesstrading,yieldingliquidityriskTheassumptionofastationaryprocess,yieldingvolatilityriskTheassumptionofcontinuoustimeandtrading,yieldinggapriskTheseshouldbeaccountedforinanyandallinvestmentstrategies,forinstancebyhedgingwithout-of-the-moneyoptions,tradingonmultipleexchanges,hedgingwithvolatilityhedgingandGammahedging,respectively.BackgroundAsbrieflymentioneditwasFischerBlackandMyronScholeswhoin1973showedthatdynamicallyrevisingaportfolioaccordingtocertainrulesremovestheexpectedreturnoftheunderlyingsecurity(Black&Scholes,1973).TheirmodelbuiltonpreviouslyestablishedworksbyBachelier,Samuelsonandothers.RobertC.Mertonwasthefirsttopublishapaperexpandingontheunderstandingofthemodelandwhocoinedtheterm“Black-Scholesoptionspricingmodel”.ScholesandMertonwasawardedthe1997NobelMemorialPrizeinEconomicSciencesfortheirdiscoveryofthemethodofdivorcingstockoptionsfromtheriskoftheirunderlyingsecurities.AsFischerBlackpassedawayin1995,hewouldnotbeeligibletoreceivetheaward,butwasacknowledgedasacontributorbytheNobelAcademy.DisclaimerIamnotamathematicaleconomist,norisanypartofthisoranyarticleIpublishmeantasfinancialadvice.Forthoseinterestedinreadingmoreaboutoptionstrading,IespeciallyrecommendthenowfamousbookTheBigShort*byMichaelLewisandperhapsalsomyownessayson“BrownianMotioninFinancialMarkets”and“Event-driveninvestments,inflectionpoints,andhowImade32xmymoneyintwoweeks”.Agood,free,Black-ScholescalculatorisavailableonWolframAlpha.Thisessayispartofaseriesofstoriesonmath-relatedtopics,publishedinCantor’sParadise,aweeklyMediumpublication.Thankyouforreading!*ThisessaycontainsAmazonAffiliatelinksCantor’sParadiseMedium’s#1MathPublication!Follow1.4K12InvestingFinanceMoneyMathEconomics1.4K claps1.4K12WrittenbyJørgenVeisdalFollowAuthorofPrivatdozent.Editor-in-ChiefatCantor’sParadise.Associateprofessor.FollowCantor’sParadiseFollowMedium’s#1MathPublicationFollowWrittenbyJørgenVeisdalFollowAuthorofPrivatdozent.Editor-in-ChiefatCantor’sParadise.Associateprofessor.Cantor’sParadiseFollowMedium’s#1MathPublicationMoreFromMediumHowtofinanceapropertypurchaseinIndiaVin.GpersonalloanhuntingtonbankAlirezadaneInsightIntoTheHistoryOfTheLendingMarket,Part1Getline.ininGetline.inHowToBudgetGroceriesWith$5Number16BusShelterOlympTrade:5CandlestickPatternsforMakeGreatProfitsEmmanuelNwankwo5thingsIwouldtellmy20-year-oldselfaboutcreditcardsBusinessInsiderinBusinessInsiderInvestInYourself,NotStocks!BillyPalla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