Black–Scholes model - Wikipedia

Black–Scholes formula Black–Scholesmodel FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Thisarticle'stoneorstylemaynotreflecttheencyclopedictoneusedonWikipedia.SeeWikipedia'sguidetowritingbetterarticlesforsuggestions.(July2020)(Learnhowandwhentoremovethistemplatemessage) Mathematicalmodeloffinancialmarkets TheBlack–Scholes/ˌblækˈʃoʊlz/[1]orBlack–Scholes–Mertonmodelisamathematicalmodelforthedynamicsofafinancialmarketcontainingderivativeinvestmentinstruments.Fromthepartialdifferentialequationinthemodel,knownastheBlack–Scholesequation,onecandeducetheBlack–Scholesformula,whichgivesatheoreticalestimateofthepriceofEuropean-styleoptionsandshowsthattheoptionhasauniquepricegiventheriskofthesecurityanditsexpectedreturn(insteadreplacingthesecurity'sexpectedreturnwiththerisk-neutralrate).TheequationandmodelarenamedaftereconomistsFischerBlackandMyronScholes;RobertC.Merton,whofirstwroteanacademicpaperonthesubject,issometimesalsocredited. Thekeyideabehindthemodelistohedgetheoptionbybuyingandsellingtheunderlyingassetinjusttherightwayand,asaconsequence,toeliminaterisk.Thistypeofhedgingiscalled"continuouslyreviseddeltahedging"andisthebasisofmorecomplicatedhedgingstrategiessuchasthoseengagedinbyinvestmentbanksandhedgefunds. Themodeliswidelyused,althoughoftenwithsomeadjustments,byoptionsmarketparticipants.[2]: 751 Themodel'sassumptionshavebeenrelaxedandgeneralizedinmanydirections,leadingtoaplethoraofmodelsthatarecurrentlyusedinderivativepricingandriskmanagement.Itistheinsightsofthemodel,asexemplifiedintheBlack–Scholesformula,thatarefrequentlyusedbymarketparticipants,asdistinguishedfromtheactualprices.Theseinsightsincludeno-arbitrageboundsandrisk-neutralpricing(thankstocontinuousrevision).Further,theBlack–Scholesequation,apartialdifferentialequationthatgovernsthepriceoftheoption,enablespricingusingnumericalmethodswhenanexplicitformulaisnotpossible. TheBlack–Scholesformulahasonlyoneparameterthatcannotbedirectlyobservedinthemarket:theaveragefuturevolatilityoftheunderlyingasset,thoughitcanbefoundfromthepriceofotheroptions.Sincetheoptionvalue(whetherputorcall)isincreasinginthisparameter,itcanbeinvertedtoproducea"volatilitysurface"thatisthenusedtocalibrateothermodels,e.g.forOTCderivatives. Contents 1History 2Fundamentalhypotheses 3Notation 4Black–Scholesequation 5Black–Scholesformula 5.1Alternativeformulation 5.2Interpretation 5.2.1Derivations 6TheOptionsGreeks 7Extensionsofthemodel 7.1Instrumentspayingcontinuousyielddividends 7.2Instrumentspayingdiscreteproportionaldividends 7.3Americanoptions 7.3.1Perpetualput 7.4Binaryoptions 7.4.1Cash-or-nothingcall 7.4.2Cash-or-nothingput 7.4.3Asset-or-nothingcall 7.4.4Asset-or-nothingput 7.4.5ForeignExchange(FX) 7.4.6Skew 7.4.7Relationshiptovanillaoptions'Greeks 8Black–Scholesinpractice 8.1Thevolatilitysmile 8.2Valuingbondoptions 8.3Interest-ratecurve 8.4Shortstockrate 9Criticismandcomments 10Seealso 11Notes 12References 12.1Primaryreferences 12.2Historicalandsociologicalaspects 12.3Furtherreading 13Externallinks 13.1Discussionofthemodel 13.2Derivationandsolution 13.3Computerimplementations 13.4Historical History[edit] EconomistsFischerBlackandMyronScholesdemonstratedin1968thatadynamicrevisionofaportfolioremovestheexpectedreturnofthesecurity,thusinventingtheriskneutralargument.[3][4]TheybasedtheirthinkingonworkpreviouslydonebymarketresearchersandpractitionersincludingLouisBachelier,SheenKassoufandEdwardO.Thorp.BlackandScholesthenattemptedtoapplytheformulatothemarkets,butincurredfinanciallosses,duetoalackofriskmanagementintheirtrades.In1970,theydecidedtoreturntotheacademicenvironment.[5]Afterthreeyearsofefforts,theformula—namedinhonorofthemformakingitpublic—wasfinallypublishedin1973inanarticletitled"ThePricingofOptionsandCorporateLiabilities",intheJournalofPoliticalEconomy.[6][7][8]RobertC.Mertonwasthefirsttopublishapaperexpandingthemathematicalunderstandingoftheoptionspricingmodel,andcoinedtheterm"Black–Scholesoptionspricingmodel". TheformulaledtoaboominoptionstradingandprovidedmathematicallegitimacytotheactivitiesoftheChicagoBoardOptionsExchangeandotheroptionsmarketsaroundtheworld.[9] MertonandScholesreceivedthe1997NobelMemorialPrizeinEconomicSciencesfortheirwork,thecommitteecitingtheirdiscoveryoftheriskneutraldynamicrevisionasabreakthroughthatseparatestheoptionfromtheriskoftheunderlyingsecurity.[10]Althoughineligiblefortheprizebecauseofhisdeathin1995,BlackwasmentionedasacontributorbytheSwedishAcademy.[11] Fundamentalhypotheses[edit] TheBlack–Scholesmodelassumesthatthemarketconsistsofatleastoneriskyasset,usuallycalledthestock,andonerisklessasset,usuallycalledthemoneymarket,cash,orbond. Nowwemakeassumptionsontheassets(whichexplaintheirnames): (Risklessrate)Therateofreturnontherisklessassetisconstantandthuscalledtherisk-freeinterestrate. (Randomwalk)Theinstantaneouslogreturnofstockpriceisaninfinitesimalrandomwalkwithdrift;moreprecisely,thestockpricefollowsageometricBrownianmotion,andwewillassumeitsdriftandvolatilityareconstant(iftheyaretime-varying,wecandeduceasuitablymodifiedBlack–Scholesformulaquitesimply,aslongasthevolatilityisnotrandom). Thestockdoesnotpayadividend.[Notes1] Theassumptionsonthemarketare: Noarbitrageopportunity(i.e.,thereisnowaytomakearisklessprofit). Abilitytoborrowandlendanyamount,evenfractional,ofcashattherisklessrate. Abilitytobuyandsellanyamount,evenfractional,ofthestock(Thisincludesshortselling). Theabovetransactionsdonotincuranyfeesorcosts(i.e.,frictionlessmarket). Withtheseassumptionsholding,supposethereisaderivativesecurityalsotradinginthismarket.Wespecifythatthissecuritywillhaveacertainpayoffataspecifieddateinthefuture,dependingonthevaluestakenbythestockuptothatdate.Itisasurprisingfactthatthederivative'spriceiscompletelydeterminedatthecurrenttime,eventhoughwedonotknowwhatpaththestockpricewilltakeinthefuture.ForthespecialcaseofaEuropeancallorputoption,BlackandScholesshowedthat"itispossibletocreateahedgedposition,consistingofalongpositioninthestockandashortpositionintheoption,whosevaluewillnotdependonthepriceofthestock".[12]Theirdynamichedgingstrategyledtoapartialdifferentialequationwhichgovernedthepriceoftheoption.ItssolutionisgivenbytheBlack–Scholesformula. Severaloftheseassumptionsoftheoriginalmodelhavebeenremovedinsubsequentextensionsofthemodel.Modernversionsaccountfordynamicinterestrates(Merton,1976),[citationneeded]transactioncostsandtaxes(Ingersoll,1976),[citationneeded]anddividendpayout.[13] Notation[edit] Thenotationusedthroughoutthispagewillbedefinedasfollows,groupedbysubject: Generalandmarketrelated: t {\displaystylet} ,atimeinyears;wegenerallyuse t = 0 {\displaystylet=0} asnow; r {\displaystyler} ,theannualizedrisk-freeinterestrate,continuouslycompoundedAlsoknownastheforceofinterest; Assetrelated: S ( t ) {\displaystyleS(t)} ,thepriceoftheunderlyingassetattimet,alsodenotedas S t {\displaystyleS_{t}} ; μ {\displaystyle\mu} ,thedriftrateof S {\displaystyleS} ,annualized; σ {\displaystyle\sigma} ,thestandarddeviationorStdofthestock'sreturns;thisisthesquarerootofthequadraticvariationofthestock'slogpriceprocess,ameasureofitsvolatility; Optionrelated: V ( S , t ) {\displaystyleV(S,t)} ,thepriceoftheoptionasafunctionoftheunderlyingassetS,attimet;inparticular C ( S , t ) {\displaystyleC(S,t)} isthepriceofaEuropeancalloptionand P ( S , t ) {\displaystyleP(S,t)} thepriceofaEuropeanputoption; T {\displaystyleT} ,timeofoptionexpiration; τ {\displaystyle\tau} ,timeuntilmaturity,whichisequalto τ = T − t {\displaystyle\tau=T-t} ; K {\displaystyleK} ,thestrikepriceoftheoption,alsoknownastheexerciseprice. Wewilluse N ( x ) {\displaystyleN(x)} todenotethestandardnormalcumulativedistributionfunction, N ( x ) = 1 2 π ∫ − ∞ x e − z 2 / 2 d z . {\displaystyleN(x)={\frac{1}{\sqrt{2\pi}}}\int_{-\infty}^{x}e^{-z^{2}/2}\,dz.} remark N ( − x ) = 1 − N ( x ) {\displaystyleN(-x)=1-N(x)} . N ′ ( x ) {\displaystyleN'(x)} willdenotethestandardnormalprobabilitydensityfunction, N ′ ( x ) = d N ( x ) d x = 1 2 π e − x 2 / 2 . {\displaystyleN'(x)={\frac{dN(x)}{dx}}={\frac{1}{\sqrt{2\pi}}}e^{-x^{2}/2}.} Black–Scholesequation[edit] Mainarticle:Black–Scholesequation SimulatedgeometricBrownianmotionswithparametersfrommarketdata TheBlack–Scholesequationisapartialdifferentialequation,whichdescribesthepriceoftheoptionovertime.Theequationis: ∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V = 0 {\displaystyle{\frac{\partialV}{\partialt}}+{\frac{1}{2}}\sigma^{2}S^{2}{\frac{\partial^{2}V}{\partialS^{2}}}+rS{\frac{\partialV}{\partialS}}-rV=0} Thekeyfinancialinsightbehindtheequationisthatonecanperfectlyhedgetheoptionbybuyingandsellingtheunderlyingassetandthebankaccountasset(cash)injusttherightwayandconsequently"eliminaterisk".[citationneeded]Thishedge,inturn,impliesthatthereisonlyonerightpricefortheoption,asreturnedbytheBlack–Scholesformula(seethenextsection). Black–Scholesformula[edit] AEuropeancallvaluedusingtheBlack–Scholespricingequationforvaryingassetprice S {\displaystyleS} andtime-to-expiry T {\displaystyleT} .Inthisparticularexample,thestrikepriceissetto1. TheBlack–ScholesformulacalculatesthepriceofEuropeanputandcalloptions.ThispriceisconsistentwiththeBlack–Scholesequationasabove;thisfollowssincetheformulacanbeobtainedbysolvingtheequationforthecorrespondingterminalandboundaryconditions: C ( 0 , t ) = 0  forall  t C ( S , t ) → S  as  S → ∞ C ( S , T ) = max { S − K , 0 } {\displaystyle{\begin{aligned}&C(0,t)=0{\text{forall}}t\\&C(S,t)\rightarrowS{\text{as}}S\rightarrow\infty\\&C(S,T)=\max\{S-K,0\}\end{aligned}}} Thevalueofacalloptionforanon-dividend-payingunderlyingstockintermsoftheBlack–Scholesparametersis: C ( S t , t ) = N ( d 1 ) S t − N ( d 2 ) K e − r ( T − t ) d 1 = 1 σ T − t [ ln ⁡ ( S t K ) + ( r + σ 2 2 ) ( T − t ) ] d 2 = d 1 − σ T − t {\displaystyle{\begin{aligned}C(S_{t},t)&=N(d_{1})S_{t}-N(d_{2})Ke^{-r(T-t)}\\d_{1}&={\frac{1}{\sigma{\sqrt{T-t}}}}\left[\ln\left({\frac{S_{t}}{K}}\right)+\left(r+{\frac{\sigma^{2}}{2}}\right)(T-t)\right]\\d_{2}&=d_{1}-\sigma{\sqrt{T-t}}\\\end{aligned}}} Thepriceofacorrespondingputoptionbasedonput–callparitywithdiscountfactor e − r ( T − t ) {\displaystylee^{-r(T-t)}} is: P ( S t , t ) = K e − r ( T − t ) − S t + C ( S t , t ) = N ( − d 2 ) K e − r ( T − t ) − N ( − d 1 ) S t {\displaystyle{\begin{aligned}P(S_{t},t)&=Ke^{-r(T-t)}-S_{t}+C(S_{t},t)\\&=N(-d_{2})Ke^{-r(T-t)}-N(-d_{1})S_{t}\end{aligned}}\,} Alternativeformulation[edit] Introducingsomeauxiliaryvariablesallowstheformulatobesimplifiedandreformulatedinaformthatisoftenmoreconvenient(thisisaspecialcaseoftheBlack'76formula): C ( F , τ ) = D [ N ( d + ) F − N ( d − ) K ] d + = 1 σ τ [ ln ⁡ ( F K ) + 1 2 σ 2 τ ] d − = d + − σ τ {\displaystyle{\begin{aligned}C(F,\tau)&=D\left[N(d_{+})F-N(d_{-})K\right]\\d_{+}&={\frac{1}{\sigma{\sqrt{\tau}}}}\left[\ln\left({\frac{F}{K}}\right)+{\frac{1}{2}}\sigma^{2}\tau\right]\\d_{-}&=d_{+}-\sigma{\sqrt{\tau}}\end{aligned}}} Theauxiliaryvariablesare: D = e − r τ {\displaystyleD=e^{-r\tau}} isthediscountfactor F = e r τ S = S D {\displaystyleF=e^{r\tau}S={\frac{S}{D}}} istheforwardpriceoftheunderlyingasset,and S = D F {\displaystyleS=DF} withd+=d1andd−=d2toclarifynotation. Givenput–callparity,whichisexpressedinthesetermsas: C − P = D ( F − K ) = S − D K {\displaystyleC-P=D(F-K)=S-DK} thepriceofaputoptionis: P ( F , τ ) = D [ N ( − d − ) K − N ( − d + ) F ] {\displaystyleP(F,\tau)=D\left[N(-d_{-})K-N(-d_{+})F\right]} Interpretation[edit] TheBlack–Scholesformulacanbeinterpretedfairlyhandily,withthemainsubtletytheinterpretationofthe N ( d ± ) {\displaystyleN(d_{\pm})} (andafortiori d ± {\displaystyled_{\pm}} )terms,particularly d + {\displaystyled_{+}} andwhytherearetwodifferentterms.[14] Theformulacanbeinterpretedbyfirstdecomposingacalloptionintothedifferenceoftwobinaryoptions:anasset-or-nothingcallminusacash-or-nothingcall(longanasset-or-nothingcall,shortacash-or-nothingcall).Acalloptionexchangescashforanassetatexpiry,whileanasset-or-nothingcalljustyieldstheasset(withnocashinexchange)andacash-or-nothingcalljustyieldscash(withnoassetinexchange).TheBlack–Scholesformulaisadifferenceoftwoterms,andthesetwotermsequalthevaluesofthebinarycalloptions.Thesebinaryoptionsaremuchlessfrequentlytradedthanvanillacalloptions,butareeasiertoanalyze. Thustheformula: C = D [ N ( d + ) F − N ( d − ) K ] {\displaystyleC=D\left[N(d_{+})F-N(d_{-})K\right]} breaksupas: C = D N ( d + ) F − D N ( d − ) K , {\displaystyleC=DN(d_{+})F-DN(d_{-})K,} where D N ( d + ) F {\displaystyleDN(d_{+})F} isthepresentvalueofanasset-or-nothingcalland D N ( d − ) K {\displaystyleDN(d_{-})K} isthepresentvalueofacash-or-nothingcall.TheDfactorisfordiscounting,becausetheexpirationdateisinfuture,andremovingitchangespresentvaluetofuturevalue(valueatexpiry).Thus N ( d + )   F {\displaystyleN(d_{+})~F} isthefuturevalueofanasset-or-nothingcalland N ( d − )   K {\displaystyleN(d_{-})~K} isthefuturevalueofacash-or-nothingcall.Inrisk-neutralterms,thesearetheexpectedvalueoftheassetandtheexpectedvalueofthecashintherisk-neutralmeasure. Thenaive,andnotquitecorrect,interpretationofthesetermsisthat N ( d + ) F {\displaystyleN(d_{+})F} istheprobabilityoftheoptionexpiringinthemoney N ( d + ) {\displaystyleN(d_{+})} ,timesthevalueoftheunderlyingatexpiryF,while N ( d − ) K {\displaystyleN(d_{-})K} istheprobabilityoftheoptionexpiringinthemoney N ( d − ) , {\displaystyleN(d_{-}),} timesthevalueofthecashatexpiryK.Thisisobviouslyincorrect,aseitherbothbinariesexpireinthemoneyorbothexpireoutofthemoney(eithercashisexchangedforassetoritisnot),buttheprobabilities N ( d + ) {\displaystyleN(d_{+})} and N ( d − ) {\displaystyleN(d_{-})} arenotequal.Infact, d ± {\displaystyled_{\pm}} canbeinterpretedasmeasuresofmoneyness(instandarddeviations)and N ( d ± ) {\displaystyleN(d_{\pm})} asprobabilitiesofexpiringITM(percentmoneyness),intherespectivenuméraire,asdiscussedbelow.Simplyput,theinterpretationofthecashoption, N ( d − ) K {\displaystyleN(d_{-})K} ,iscorrect,asthevalueofthecashisindependentofmovementsoftheunderlyingasset,andthuscanbeinterpretedasasimpleproductof"probabilitytimesvalue",whilethe N ( d + ) F {\displaystyleN(d_{+})F} ismorecomplicated,astheprobabilityofexpiringinthemoneyandthevalueoftheassetatexpiryarenotindependent.[14]Moreprecisely,thevalueoftheassetatexpiryisvariableintermsofcash,butisconstantintermsoftheassetitself(afixedquantityoftheasset),andthusthesequantitiesareindependentifonechangesnumérairetotheassetratherthancash. IfoneusesspotSinsteadofforwardF,in d ± {\displaystyled_{\pm}} insteadofthe 1 2 σ 2 {\textstyle{\frac{1}{2}}\sigma^{2}} termthereis ( r ± 1 2 σ 2 ) τ , {\textstyle\left(r\pm{\frac{1}{2}}\sigma^{2}\right)\tau,} whichcanbeinterpretedasadriftfactor(intherisk-neutralmeasureforappropriatenuméraire).Theuseofd−formoneynessratherthanthestandardizedmoneyness m = 1 σ τ ln ⁡ ( F K ) {\textstylem={\frac{1}{\sigma{\sqrt{\tau}}}}\ln\left({\frac{F}{K}}\right)}  –inotherwords,thereasonforthe 1 2 σ 2 {\textstyle{\frac{1}{2}}\sigma^{2}} factor –isduetothedifferencebetweenthemedianandmeanofthelog-normaldistribution;itisthesamefactorasinItō'slemmaappliedtogeometricBrownianmotion.Inaddition,anotherwaytoseethatthenaiveinterpretationisincorrectisthatreplacing N ( d + ) {\displaystyleN(d_{+})} by N ( d − ) {\displaystyleN(d_{-})} intheformulayieldsanegativevalueforout-of-the-moneycalloptions.[14]: 6  Indetail,theterms N ( d 1 ) , N ( d 2 ) {\displaystyleN(d_{1}),N(d_{2})} aretheprobabilitiesoftheoptionexpiringin-the-moneyundertheequivalentexponentialmartingaleprobabilitymeasure(numéraire=stock)andtheequivalentmartingaleprobabilitymeasure(numéraire=riskfreeasset),respectively.[14]Theriskneutralprobabilitydensityforthestockprice S T ∈ ( 0 , ∞ ) {\displaystyleS_{T}\in(0,\infty)} is p ( S , T ) = N ′ [ d 2 ( S T ) ] S T σ T {\displaystylep(S,T)={\frac{N^{\prime}[d_{2}(S_{T})]}{S_{T}\sigma{\sqrt{T}}}}} where d 2 = d 2 ( K ) {\displaystyled_{2}=d_{2}(K)} isdefinedasabove. Specifically, N ( d 2 ) {\displaystyleN(d_{2})} istheprobabilitythatthecallwillbeexercisedprovidedoneassumesthattheassetdriftistherisk-freerate. N ( d 1 ) {\displaystyleN(d_{1})} ,however,doesnotlenditselftoasimpleprobabilityinterpretation. S N ( d 1 ) {\displaystyleSN(d_{1})} iscorrectlyinterpretedasthepresentvalue,usingtherisk-freeinterestrate,oftheexpectedassetpriceatexpiration,giventhattheassetpriceatexpirationisabovetheexerciseprice.[15]Forrelateddiscussion –andgraphicalrepresentation –seeDatar–Mathewsmethodforrealoptionvaluation. Theequivalentmartingaleprobabilitymeasureisalsocalledtherisk-neutralprobabilitymeasure.Notethatbothoftheseareprobabilitiesinameasuretheoreticsense,andneitheroftheseisthetrueprobabilityofexpiringin-the-moneyundertherealprobabilitymeasure.Tocalculatetheprobabilityunderthereal("physical")probabilitymeasure,additionalinformationisrequired—thedriftterminthephysicalmeasure,orequivalently,themarketpriceofrisk. Derivations[edit] Seealso:Martingalepricing AstandardderivationforsolvingtheBlack–ScholesPDEisgiveninthearticleBlack–Scholesequation. TheFeynman–KacformulasaysthatthesolutiontothistypeofPDE,whendiscountedappropriately,isactuallyamartingale.Thustheoptionpriceistheexpectedvalueofthediscountedpayoffoftheoption.ComputingtheoptionpriceviathisexpectationistheriskneutralityapproachandcanbedonewithoutknowledgeofPDEs.[14]Notetheexpectationoftheoptionpayoffisnotdoneundertherealworldprobabilitymeasure,butanartificialrisk-neutralmeasure,whichdiffersfromtherealworldmeasure.Fortheunderlyinglogicseesection"riskneutralvaluation"underRationalpricingaswellassection"Derivativespricing:theQworld"underMathematicalfinance;fordetails,onceagain,seeHull.[16]: 307–309  TheOptionsGreeks[edit] "TheGreeks"measurethesensitivityofthevalueofaderivativeproductorafinancialportfoliotochangesinparametervalueswhileholdingtheotherparametersfixed.Theyarepartialderivativesofthepricewithrespecttotheparametervalues.OneGreek,"gamma"(aswellasothersnotlistedhere)isapartialderivativeofanotherGreek,"delta"inthiscase. TheGreeksareimportantnotonlyinthemathematicaltheoryoffinance,butalsoforthoseactivelytrading.Financialinstitutionswilltypicallyset(risk)limitvaluesforeachoftheGreeksthattheirtradersmustnotexceed.DeltaisthemostimportantGreeksincethisusuallyconfersthelargestrisk.Manytraderswillzerotheirdeltaattheendofthedayiftheyarenotspeculatingonthedirectionofthemarketandfollowingadelta-neutralhedgingapproachasdefinedbyBlack–Scholes. TheGreeksforBlack–Scholesaregiveninclosedformbelow.TheycanbeobtainedbydifferentiationoftheBlack–Scholesformula.[17] Call Put Delta ∂ V ∂ S {\displaystyle{\frac{\partialV}{\partialS}}} N ( d 1 ) {\displaystyleN(d_{1})\,} − N ( − d 1 ) = N ( d 1 ) − 1 {\displaystyle-N(-d_{1})=N(d_{1})-1\,} Gamma ∂ 2 V ∂ S 2 {\displaystyle{\frac{\partial^{2}V}{\partialS^{2}}}} N ′ ( d 1 ) S σ T − t {\displaystyle{\frac{N'(d_{1})}{S\sigma{\sqrt{T-t}}}}\,} Vega ∂ V ∂ σ {\displaystyle{\frac{\partialV}{\partial\sigma}}} S N ′ ( d 1 ) T − t {\displaystyleSN'(d_{1}){\sqrt{T-t}}\,} Theta ∂ V ∂ t {\displaystyle{\frac{\partialV}{\partialt}}} − S N ′ ( d 1 ) σ 2 T − t − r K e − r ( T − t ) N ( d 2 ) {\displaystyle-{\frac{SN'(d_{1})\sigma}{2{\sqrt{T-t}}}}-rKe^{-r(T-t)}N(d_{2})\,} − S N ′ ( d 1 ) σ 2 T − t + r K e − r ( T − t ) N ( − d 2 ) {\displaystyle-{\frac{SN'(d_{1})\sigma}{2{\sqrt{T-t}}}}+rKe^{-r(T-t)}N(-d_{2})\,} Rho ∂ V ∂ r {\displaystyle{\frac{\partialV}{\partialr}}} K ( T − t ) e − r ( T − t ) N ( d 2 ) {\displaystyleK(T-t)e^{-r(T-t)}N(d_{2})\,} − K ( T − t ) e − r ( T − t ) N ( − d 2 ) {\displaystyle-K(T-t)e^{-r(T-t)}N(-d_{2})\,} Notethatfromtheformulae,itisclearthatthegammaisthesamevalueforcallsandputsandsotooisthevegathesamevalueforcallsandputsoptions.Thiscanbeseendirectlyfromput–callparity,sincethedifferenceofaputandacallisaforward,whichislinearinSandindependentofσ(soaforwardhaszerogammaandzerovega).N'isthestandardnormalprobabilitydensityfunction. Inpractice,somesensitivitiesareusuallyquotedinscaled-downterms,tomatchthescaleoflikelychangesintheparameters.Forexample,rhoisoftenreporteddividedby10,000(1basispointratechange),vegaby100(1volpointchange),andthetaby365or252(1daydecaybasedoneithercalendardaysortradingdaysperyear). Notethat"Vega"isnotaletterintheGreekalphabet;thenamearisesfrommisreadingtheGreekletternu(variouslyrenderedas ν {\displaystyle\nu} ,ν,andν)asaV. Extensionsofthemodel[edit] Theabovemodelcanbeextendedforvariable(butdeterministic)ratesandvolatilities.ThemodelmayalsobeusedtovalueEuropeanoptionsoninstrumentspayingdividends.Inthiscase,closed-formsolutionsareavailableifthedividendisaknownproportionofthestockprice.Americanoptionsandoptionsonstockspayingaknowncashdividend(intheshortterm,morerealisticthanaproportionaldividend)aremoredifficulttovalue,andachoiceofsolutiontechniquesisavailable(forexamplelatticesandgrids). Instrumentspayingcontinuousyielddividends[edit] Foroptionsonindices,itisreasonabletomakethesimplifyingassumptionthatdividendsarepaidcontinuously,andthatthedividendamountisproportionaltotheleveloftheindex. Thedividendpaymentpaidoverthetimeperiod [ t , t + d t ] {\displaystyle[t,t+dt]} isthenmodelledas : q S t d t {\displaystyleqS_{t}\,dt} forsomeconstant q {\displaystyleq} (thedividendyield). Underthisformulationthearbitrage-freepriceimpliedbytheBlack–Scholesmodelcanbeshowntobe : C ( S t , t ) = e − r ( T − t ) [ F N ( d 1 ) − K N ( d 2 ) ] {\displaystyleC(S_{t},t)=e^{-r(T-t)}[FN(d_{1})-KN(d_{2})]\,} and P ( S t , t ) = e − r ( T − t ) [ K N ( − d 2 ) − F N ( − d 1 ) ] {\displaystyleP(S_{t},t)=e^{-r(T-t)}[KN(-d_{2})-FN(-d_{1})]\,} wherenow F = S t e ( r − q ) ( T − t ) {\displaystyleF=S_{t}e^{(r-q)(T-t)}\,} isthemodifiedforwardpricethatoccursintheterms d 1 , d 2 {\displaystyled_{1},d_{2}} : d 1 = 1 σ T − t [ ln ⁡ ( S t K ) + ( r − q + 1 2 σ 2 ) ( T − t ) ] {\displaystyled_{1}={\frac{1}{\sigma{\sqrt{T-t}}}}\left[\ln\left({\frac{S_{t}}{K}}\right)+\left(r-q+{\frac{1}{2}}\sigma^{2}\right)(T-t)\right]} and d 2 = d 1 − σ T − t = 1 σ T − t [ ln ⁡ ( S t K ) + ( r − q − 1 2 σ 2 ) ( T − t ) ] {\displaystyled_{2}=d_{1}-\sigma{\sqrt{T-t}}={\frac{1}{\sigma{\sqrt{T-t}}}}\left[\ln\left({\frac{S_{t}}{K}}\right)+\left(r-q-{\frac{1}{2}}\sigma^{2}\right)(T-t)\right]} .[18] Instrumentspayingdiscreteproportionaldividends[edit] ItisalsopossibletoextendtheBlack–Scholesframeworktooptionsoninstrumentspayingdiscreteproportionaldividends.Thisisusefulwhentheoptionisstruckonasinglestock. Atypicalmodelistoassumethataproportion δ {\displaystyle\delta} ofthestockpriceispaidoutatpre-determinedtimes t 1 , t 2 , … , t n {\displaystylet_{1},t_{2},\ldots,t_{n}} .Thepriceofthestockisthenmodelledas : S t = S 0 ( 1 − δ ) n ( t ) e u t + σ W t {\displaystyleS_{t}=S_{0}(1-\delta)^{n(t)}e^{ut+\sigmaW_{t}}} where n ( t ) {\displaystylen(t)} isthenumberofdividendsthathavebeenpaidbytime t {\displaystylet} . Thepriceofacalloptiononsuchastockisagain : C ( S 0 , T ) = e − r T [ F N ( d 1 ) − K N ( d 2 ) ] {\displaystyleC(S_{0},T)=e^{-rT}[FN(d_{1})-KN(d_{2})]\,} wherenow F = S 0 ( 1 − δ ) n ( T ) e r T {\displaystyleF=S_{0}(1-\delta)^{n(T)}e^{rT}\,} istheforwardpriceforthedividendpayingstock. Americanoptions[edit] TheproblemoffindingthepriceofanAmericanoptionisrelatedtotheoptimalstoppingproblemoffindingthetimetoexecutetheoption.SincetheAmericanoptioncanbeexercisedatanytimebeforetheexpirationdate,theBlack–Scholesequationbecomesavariationalinequalityoftheform ∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + r S ∂ V ∂ S − r V ≤ 0 {\displaystyle{\frac{\partialV}{\partialt}}+{\frac{1}{2}}\sigma^{2}S^{2}{\frac{\partial^{2}V}{\partialS^{2}}}+rS{\frac{\partialV}{\partialS}}-rV\leq0} [19] togetherwith V ( S , t ) ≥ H ( S ) {\displaystyleV(S,t)\geqH(S)} where H ( S ) {\displaystyleH(S)} denotesthepayoffatstockprice S {\displaystyleS} andtheterminalcondition: V ( S , T ) = H ( S ) {\displaystyleV(S,T)=H(S)} . Ingeneralthisinequalitydoesnothaveaclosedformsolution,thoughanAmericancallwithnodividendsisequaltoaEuropeancallandtheRoll–Geske–WhaleymethodprovidesasolutionforanAmericancallwithonedividend;[20][21]seealsoBlack'sapproximation. Barone-AdesiandWhaley[22]isafurtherapproximationformula.Here,thestochasticdifferentialequation(whichisvalidforthevalueofanyderivative)issplitintotwocomponents:theEuropeanoptionvalueandtheearlyexercisepremium.Withsomeassumptions,aquadraticequationthatapproximatesthesolutionforthelatteristhenobtained.Thissolutioninvolvesfindingthecriticalvalue, s ∗ {\displaystyles*} ,suchthatoneisindifferentbetweenearlyexerciseandholdingtomaturity.[23][24] BjerksundandStensland[25]provideanapproximationbasedonanexercisestrategycorrespondingtoatriggerprice.Here,iftheunderlyingassetpriceisgreaterthanorequaltothetriggerpriceitisoptimaltoexercise,andthevaluemustequal S − X {\displaystyleS-X} ,otherwisetheoption"boilsdownto:(i)aEuropeanup-and-outcalloption…and(ii)arebatethatisreceivedattheknock-outdateiftheoptionisknockedoutpriortothematuritydate".Theformulaisreadilymodifiedforthevaluationofaputoption,usingput–callparity.Thisapproximationiscomputationallyinexpensiveandthemethodisfast,withevidenceindicatingthattheapproximationmaybemoreaccurateinpricinglongdatedoptionsthanBarone-AdesiandWhaley.[26] Perpetualput[edit] DespitethelackofageneralanalyticalsolutionforAmericanputoptions,itispossibletoderivesuchaformulaforthecaseofaperpetualoption-meaningthattheoptionneverexpires(i.e., T → ∞ {\displaystyleT\rightarrow\infty} ).[27]Inthiscase,thetimedecayoftheoptionisequaltozero,whichleadstotheBlack–ScholesPDEbecominganODE: 1 2 σ 2 S 2 d 2 V d S 2 + ( r − q ) S d V d S − r V = 0 {\displaystyle{1\over{2}}\sigma^{2}S^{2}{d^{2}V\over{dS^{2}}}+(r-q)S{dV\over{dS}}-rV=0} Let S − {\displaystyleS_{-}} denotethelowerexerciseboundary,belowwhichisoptimalforexercisingtheoption.Theboundaryconditionsare: V ( S − ) = K − S − , V S ( S − ) = − 1 , V ( S ) ≤ K {\displaystyleV(S_{-})=K-S_{-},\quadV_{S}(S_{-})=-1,\quadV(S)\leqK} ThesolutionstotheODEarealinearcombinationofanytwolinearlyindependentsolutions: V ( S ) = A 1 S λ 1 + A 2 S λ 2 {\displaystyleV(S)=A_{1}S^{\lambda_{1}}+A_{2}S^{\lambda_{2}}} For S − ≤ S {\displaystyleS_{-}\leqS} ,substitutionofthissolutionintotheODEfor i = 1 , 2 {\displaystylei={1,2}} yields: [ 1 2 σ 2 λ i ( λ i − 1 ) + ( r − q ) λ i − r ] S λ i = 0 {\displaystyle\left[{1\over{2}}\sigma^{2}\lambda_{i}(\lambda_{i}-1)+(r-q)\lambda_{i}-r\right]S^{\lambda_{i}}=0} Rearrangingthetermsingives: 1 2 σ 2 λ i 2 + ( r − q − 1 2 σ 2 ) λ i − r = 0 {\displaystyle{1\over{2}}\sigma^{2}\lambda_{i}^{2}+\left(r-q-{1\over{2}}\sigma^{2}\right)\lambda_{i}-r=0} Usingthequadraticformula,thesolutionsfor λ i {\displaystyle\lambda_{i}} are: λ 1 = − ( r − q − 1 2 σ 2 ) + ( r − q − 1 2 σ 2 ) 2 + 2 σ 2 r σ 2 λ 2 = − ( r − q − 1 2 σ 2 ) − ( r − q − 1 2 σ 2 ) 2 + 2 σ 2 r σ 2 {\displaystyle{\begin{aligned}\lambda_{1}&={-\left(r-q-{1\over{2}}\sigma^{2}\right)+{\sqrt{\left(r-q-{1\over{2}}\sigma^{2}\right)^{2}+2\sigma^{2}r}}\over{\sigma^{2}}}\\\lambda_{2}&={-\left(r-q-{1\over{2}}\sigma^{2}\right)-{\sqrt{\left(r-q-{1\over{2}}\sigma^{2}\right)^{2}+2\sigma^{2}r}}\over{\sigma^{2}}}\end{aligned}}} Inordertohaveafinitesolutionfortheperpetualput,sincetheboundaryconditionsimplyupperandlowerfiniteboundsonthevalueoftheput,itisnecessarytoset A 1 = 0 {\displaystyleA_{1}=0} ,leadingtothesolution V ( S ) = A 2 S λ 2 {\displaystyleV(S)=A_{2}S^{\lambda_{2}}} .Fromthefirstboundarycondition,itisknownthat: V ( S − ) = A 2 ( S − ) λ 2 = K − S − ⟹ A 2 = K − S − ( S − ) λ 2 {\displaystyleV(S_{-})=A_{2}(S_{-})^{\lambda_{2}}=K-S_{-}\impliesA_{2}={K-S_{-}\over{(S_{-})^{\lambda_{2}}}}} Therefore,thevalueoftheperpetualputbecomes: V ( S ) = ( K − S − ) ( S S − ) λ 2 {\displaystyleV(S)=(K-S_{-})\left({S\over{S_{-}}}\right)^{\lambda_{2}}} Thesecondboundaryconditionyieldsthelocationofthelowerexerciseboundary: V S ( S − ) = λ 2 K − S − S − = − 1 ⟹ S − = λ 2 K λ 2 − 1 {\displaystyleV_{S}(S_{-})=\lambda_{2}{K-S_{-}\over{S_{-}}}=-1\impliesS_{-}={\lambda_{2}K\over{\lambda_{2}-1}}} Toconclude,for S ≥ S − = λ 2 K λ 2 − 1 {\textstyleS\geqS_{-}={\lambda_{2}K\over{\lambda_{2}-1}}} ,theperpetualAmericanputoptionisworth: V ( S ) = K 1 − λ 2 ( λ 2 − 1 λ 2 ) λ 2 ( S K ) λ 2 {\displaystyleV(S)={K\over{1-\lambda_{2}}}\left({\lambda_{2}-1\over{\lambda_{2}}}\right)^{\lambda_{2}}\left({S\over{K}}\right)^{\lambda_{2}}} Binaryoptions[edit] BysolvingtheBlack–Scholesdifferentialequation,withforboundaryconditiontheHeavisidefunction,weendupwiththepricingofoptionsthatpayoneunitabovesomepredefinedstrikepriceandnothingbelow.[28] Infact,theBlack–Scholesformulaforthepriceofavanillacalloption(orputoption)canbeinterpretedbydecomposingacalloptionintoanasset-or-nothingcalloptionminusacash-or-nothingcalloption,andsimilarlyforaput–thebinaryoptionsareeasiertoanalyze,andcorrespondtothetwotermsintheBlack–Scholesformula. Cash-or-nothingcall[edit] Thispaysoutoneunitofcashifthespotisabovethestrikeatmaturity.Itsvalueisgivenby : C = e − r ( T − t ) N ( d 2 ) . {\displaystyleC=e^{-r(T-t)}N(d_{2}).\,} Cash-or-nothingput[edit] Thispaysoutoneunitofcashifthespotisbelowthestrikeatmaturity.Itsvalueisgivenby : P = e − r ( T − t ) N ( − d 2 ) . {\displaystyleP=e^{-r(T-t)}N(-d_{2}).\,} Asset-or-nothingcall[edit] Thispaysoutoneunitofassetifthespotisabovethestrikeatmaturity.Itsvalueisgivenby : C = S e − q ( T − t ) N ( d 1 ) . {\displaystyleC=Se^{-q(T-t)}N(d_{1}).\,} Asset-or-nothingput[edit] Thispaysoutoneunitofassetifthespotisbelowthestrikeatmaturity.Itsvalueisgivenby : P = S e − q ( T − t ) N ( − d 1 ) , {\displaystyleP=Se^{-q(T-t)}N(-d_{1}),} ForeignExchange(FX)[edit] Furtherinformation:Foreignexchangederivative IfwedenotebyStheFOR/DOMexchangerate(i.e.,1unitofforeigncurrencyisworthSunitsofdomesticcurrency)wecanobservethatpayingout1unitofthedomesticcurrencyifthespotatmaturityisaboveorbelowthestrikeisexactlylikeacash-ornothingcallandputrespectively.Similarly,payingout1unitoftheforeigncurrencyifthespotatmaturityisaboveorbelowthestrikeisexactlylikeanasset-ornothingcallandputrespectively. Henceifwenowtake r F O R {\displaystyler_{FOR}} ,theforeigninterestrate, r D O M {\displaystyler_{DOM}} ,thedomesticinterestrate,andtherestasabove,wegetthefollowingresults. Incaseofadigitalcall(thisisacallFOR/putDOM)payingoutoneunitofthedomesticcurrencywegetaspresentvalue, C = e − r D O M T N ( d 2 ) {\displaystyleC=e^{-r_{DOM}T}N(d_{2})\,} Incaseofadigitalput(thisisaputFOR/callDOM)payingoutoneunitofthedomesticcurrencywegetaspresentvalue, P = e − r D O M T N ( − d 2 ) {\displaystyleP=e^{-r_{DOM}T}N(-d_{2})\,} Whileincaseofadigitalcall(thisisacallFOR/putDOM)payingoutoneunitoftheforeigncurrencywegetaspresentvalue, C = S e − r F O R T N ( d 1 ) {\displaystyleC=Se^{-r_{FOR}T}N(d_{1})\,} andincaseofadigitalput(thisisaputFOR/callDOM)payingoutoneunitoftheforeigncurrencywegetaspresentvalue, P = S e − r F O R T N ( − d 1 ) {\displaystyleP=Se^{-r_{FOR}T}N(-d_{1})\,} Skew[edit] InthestandardBlack–Scholesmodel,onecaninterpretthepremiumofthebinaryoptionintherisk-neutralworldastheexpectedvalue=probabilityofbeingin-the-money*unit,discountedtothepresentvalue.TheBlack–Scholesmodelreliesonsymmetryofdistributionandignorestheskewnessofthedistributionoftheasset.Marketmakersadjustforsuchskewnessby,insteadofusingasinglestandarddeviationfortheunderlyingasset σ {\displaystyle\sigma} acrossallstrikes,incorporatingavariableone σ ( K ) {\displaystyle\sigma(K)} wherevolatilitydependsonstrikeprice,thusincorporatingthevolatilityskewintoaccount.Theskewmattersbecauseitaffectsthebinaryconsiderablymorethantheregularoptions. Abinarycalloptionis,atlongexpirations,similartoatightcallspreadusingtwovanillaoptions.Onecanmodelthevalueofabinarycash-or-nothingoption,C,atstrikeK,asaninfinitesimallytightspread,where C v {\displaystyleC_{v}} isavanillaEuropeancall:[29][30] C = lim ϵ → 0 C v ( K − ϵ ) − C v ( K ) ϵ {\displaystyleC=\lim_{\epsilon\to0}{\frac{C_{v}(K-\epsilon)-C_{v}(K)}{\epsilon}}} Thus,thevalueofabinarycallisthenegativeofthederivativeofthepriceofavanillacallwithrespecttostrikeprice: C = − d C v d K {\displaystyleC=-{\frac{dC_{v}}{dK}}} Whenonetakesvolatilityskewintoaccount, σ {\displaystyle\sigma} isafunctionof K {\displaystyleK} : C = − d C v ( K , σ ( K ) ) d K = − ∂ C v ∂ K − ∂ C v ∂ σ ∂ σ ∂ K {\displaystyleC=-{\frac{dC_{v}(K,\sigma(K))}{dK}}=-{\frac{\partialC_{v}}{\partialK}}-{\frac{\partialC_{v}}{\partial\sigma}}{\frac{\partial\sigma}{\partialK}}} Thefirsttermisequaltothepremiumofthebinaryoptionignoringskew: − ∂ C v ∂ K = − ∂ ( S N ( d 1 ) − K e − r ( T − t ) N ( d 2 ) ) ∂ K = e − r ( T − t ) N ( d 2 ) = C noskew {\displaystyle-{\frac{\partialC_{v}}{\partialK}}=-{\frac{\partial(SN(d_{1})-Ke^{-r(T-t)}N(d_{2}))}{\partialK}}=e^{-r(T-t)}N(d_{2})=C_{\text{noskew}}} ∂ C v ∂ σ {\displaystyle{\frac{\partialC_{v}}{\partial\sigma}}} istheVegaofthevanillacall; ∂ σ ∂ K {\displaystyle{\frac{\partial\sigma}{\partialK}}} issometimescalledthe"skewslope"orjust"skew".Iftheskewistypicallynegative,thevalueofabinarycallwillbehigherwhentakingskewintoaccount. C = C noskew − Vega v ⋅ Skew {\displaystyleC=C_{\text{noskew}}-{\text{Vega}}_{v}\cdot{\text{Skew}}} Relationshiptovanillaoptions'Greeks[edit] Sinceabinarycallisamathematicalderivativeofavanillacallwithrespecttostrike,thepriceofabinarycallhasthesameshapeasthedeltaofavanillacall,andthedeltaofabinarycallhasthesameshapeasthegammaofavanillacall. Black–Scholesinpractice[edit] ThenormalityassumptionoftheBlack–Scholesmodeldoesnotcaptureextrememovementssuchasstockmarketcrashes. TheassumptionsoftheBlack–Scholesmodelarenotallempiricallyvalid.Themodeliswidelyemployedasausefulapproximationtoreality,butproperapplicationrequiresunderstandingitslimitations –blindlyfollowingthemodelexposestheusertounexpectedrisk.[31][unreliablesource?] Amongthemostsignificantlimitationsare: theunderestimationofextrememoves,yieldingtailrisk,whichcanbehedgedwithout-of-the-moneyoptions; theassumptionofinstant,cost-lesstrading,yieldingliquidityrisk,whichisdifficulttohedge; theassumptionofastationaryprocess,yieldingvolatilityrisk,whichcanbehedgedwithvolatilityhedging; theassumptionofcontinuoustimeandcontinuoustrading,yieldinggaprisk,whichcanbehedgedwithGammahedging. Inshort,whileintheBlack–ScholesmodelonecanperfectlyhedgeoptionsbysimplyDeltahedging,inpracticetherearemanyothersourcesofrisk. ResultsusingtheBlack–Scholesmodeldifferfromrealworldpricesbecauseofsimplifyingassumptionsofthemodel.Onesignificantlimitationisthatinrealitysecuritypricesdonotfollowastrictstationarylog-normalprocess,noristherisk-freeinterestactuallyknown(andisnotconstantovertime).Thevariancehasbeenobservedtobenon-constantleadingtomodelssuchasGARCHtomodelvolatilitychanges.PricingdiscrepanciesbetweenempiricalandtheBlack–Scholesmodelhavelongbeenobservedinoptionsthatarefarout-of-the-money,correspondingtoextremepricechanges;sucheventswouldbeveryrareifreturnswerelognormallydistributed,butareobservedmuchmoreofteninpractice. Nevertheless,Black–Scholespricingiswidelyusedinpractice,[2]: 751 [32]becauseitis: easytocalculate ausefulapproximation,particularlywhenanalyzingthedirectioninwhichpricesmovewhencrossingcriticalpoints arobustbasisformorerefinedmodels reversible,asthemodel'soriginaloutput,price,canbeusedasaninputandoneoftheothervariablessolvedfor;theimpliedvolatilitycalculatedinthiswayisoftenusedtoquoteoptionprices(thatis,asaquotingconvention). Thefirstpointisself-evidentlyuseful.Theotherscanbefurtherdiscussed: Usefulapproximation:althoughvolatilityisnotconstant,resultsfromthemodelareoftenhelpfulinsettinguphedgesinthecorrectproportionstominimizerisk.Evenwhentheresultsarenotcompletelyaccurate,theyserveasafirstapproximationtowhichadjustmentscanbemade. Basisformorerefinedmodels:TheBlack–Scholesmodelisrobustinthatitcanbeadjustedtodealwithsomeofitsfailures.Ratherthanconsideringsomeparameters(suchasvolatilityorinterestrates)asconstant,oneconsidersthemasvariables,andthusaddedsourcesofrisk.ThisisreflectedintheGreeks(thechangeinoptionvalueforachangeintheseparameters,orequivalentlythepartialderivativeswithrespecttothesevariables),andhedgingtheseGreeksmitigatestheriskcausedbythenon-constantnatureoftheseparameters.Otherdefectscannotbemitigatedbymodifyingthemodel,however,notablytailriskandliquidityrisk,andtheseareinsteadmanagedoutsidethemodel,chieflybyminimizingtheserisksandbystresstesting. Explicitmodeling:thisfeaturemeansthat,ratherthanassumingavolatilityaprioriandcomputingpricesfromit,onecanusethemodeltosolveforvolatility,whichgivestheimpliedvolatilityofanoptionatgivenprices,durationsandexerciseprices.Solvingforvolatilityoveragivensetofdurationsandstrikeprices,onecanconstructanimpliedvolatilitysurface.InthisapplicationoftheBlack–Scholesmodel,acoordinatetransformationfromthepricedomaintothevolatilitydomainisobtained.Ratherthanquotingoptionpricesintermsofdollarsperunit(whicharehardtocompareacrossstrikes,durationsandcouponfrequencies),optionpricescanthusbequotedintermsofimpliedvolatility,whichleadstotradingofvolatilityinoptionmarkets. Thevolatilitysmile[edit] Mainarticle:Volatilitysmile OneoftheattractivefeaturesoftheBlack–Scholesmodelisthattheparametersinthemodelotherthanthevolatility(thetimetomaturity,thestrike,therisk-freeinterestrate,andthecurrentunderlyingprice)areunequivocallyobservable.Allotherthingsbeingequal,anoption'stheoreticalvalueisamonotonicincreasingfunctionofimpliedvolatility. Bycomputingtheimpliedvolatilityfortradedoptionswithdifferentstrikesandmaturities,theBlack–Scholesmodelcanbetested.IftheBlack–Scholesmodelheld,thentheimpliedvolatilityforaparticularstockwouldbethesameforallstrikesandmaturities.Inpractice,thevolatilitysurface(the3Dgraphofimpliedvolatilityagainststrikeandmaturity)isnotflat. Thetypicalshapeoftheimpliedvolatilitycurveforagivenmaturitydependsontheunderlyinginstrument.Equitiestendtohaveskewedcurves:comparedtoat-the-money,impliedvolatilityissubstantiallyhigherforlowstrikes,andslightlylowerforhighstrikes.Currenciestendtohavemoresymmetricalcurves,withimpliedvolatilitylowestat-the-money,andhighervolatilitiesinbothwings.Commoditiesoftenhavethereversebehaviortoequities,withhigherimpliedvolatilityforhigherstrikes. Despitetheexistenceofthevolatilitysmile(andtheviolationofalltheotherassumptionsoftheBlack–Scholesmodel),theBlack–ScholesPDEandBlack–Scholesformulaarestillusedextensivelyinpractice.Atypicalapproachistoregardthevolatilitysurfaceasafactaboutthemarket,anduseanimpliedvolatilityfromitinaBlack–Scholesvaluationmodel.Thishasbeendescribedasusing"thewrongnumberinthewrongformulatogettherightprice".[33]Thisapproachalsogivesusablevaluesforthehedgeratios(theGreeks).Evenwhenmoreadvancedmodelsareused,tradersprefertothinkintermsofBlack–Scholesimpliedvolatilityasitallowsthemtoevaluateandcompareoptionsofdifferentmaturities,strikes,andsoon.Foradiscussionastothevariousalternativeapproachesdevelopedhere,seeFinancialeconomics§ Challengesandcriticism. Valuingbondoptions[edit] Black–Scholescannotbeapplieddirectlytobondsecuritiesbecauseofpull-to-par.Asthebondreachesitsmaturitydate,allofthepricesinvolvedwiththebondbecomeknown,therebydecreasingitsvolatility,andthesimpleBlack–Scholesmodeldoesnotreflectthisprocess.AlargenumberofextensionstoBlack–Scholes,beginningwiththeBlackmodel,havebeenusedtodealwiththisphenomenon.[34]SeeBondoption§ Valuation. Interest-ratecurve[edit] Inpractice,interestratesarenotconstant –theyvarybytenor(couponfrequency),givinganinterestratecurvewhichmaybeinterpolatedtopickanappropriateratetouseintheBlack–Scholesformula.Anotherconsiderationisthatinterestratesvaryovertime.Thisvolatilitymaymakeasignificantcontributiontotheprice,especiallyoflong-datedoptions.Thisissimplyliketheinterestrateandbondpricerelationshipwhichisinverselyrelated. Shortstockrate[edit] Takingashortstockposition,asinherentinthederivation,isnottypicallyfreeofcost;equivalently,itispossibletolendoutalongstockpositionforasmallfee.Ineithercase,thiscanbetreatedasacontinuousdividendforthepurposesofaBlack–Scholesvaluation,providedthatthereisnoglaringasymmetrybetweentheshortstockborrowingcostandthelongstocklendingincome.[citationneeded] Criticismandcomments[edit] EspenGaarderHaugandNassimNicholasTalebarguethattheBlack–Scholesmodelmerelyrecastsexistingwidelyusedmodelsintermsofpracticallyimpossible"dynamichedging"ratherthan"risk",tomakethemmorecompatiblewithmainstreamneoclassicaleconomictheory.[35]TheyalsoassertthatBonessin1964hadalreadypublishedaformulathatis"actuallyidentical"totheBlack–Scholescalloptionpricingequation.[36]EdwardThorpalsoclaimstohaveguessedtheBlack–Scholesformulain1967butkeptittohimselftomakemoneyforhisinvestors.[37]EmanuelDermanandNassimTalebhavealsocriticizeddynamichedgingandstatethatanumberofresearchershadputforthsimilarmodelspriortoBlackandScholes.[38]Inresponse,PaulWilmotthasdefendedthemodel.[32][39] Inhis2008lettertotheshareholdersofBerkshireHathaway,WarrenBuffettwrote:"IbelievetheBlack–Scholesformula,eventhoughitisthestandardforestablishingthedollarliabilityforoptions,producesstrangeresultswhenthelong-termvarietyarebeingvalued...TheBlack–Scholesformulahasapproachedthestatusofholywritinfinance...Iftheformulaisappliedtoextendedtimeperiods,however,itcanproduceabsurdresults.Infairness,BlackandScholesalmostcertainlyunderstoodthispointwell.Buttheirdevotedfollowersmaybeignoringwhatevercaveatsthetwomenattachedwhentheyfirstunveiledtheformula."[40] BritishmathematicianIanStewart,authorofthe2012bookentitledInPursuitoftheUnknown:17EquationsThatChangedtheWorld,[41][42]saidthatBlack–Scholeshad"underpinnedmassiveeconomicgrowth"andthe"internationalfinancialsystemwastradingderivativesvaluedatonequadrilliondollarsperyear"by2007.HesaidthattheBlack–Scholesequationwasthe"mathematicaljustificationforthetrading"—andtherefore—"oneingredientinarichstewoffinancialirresponsibility,politicalineptitude,perverseincentivesandlaxregulation"thatcontributedtothefinancialcrisisof2007–08.[43]Heclarifiedthat"theequationitselfwasn'ttherealproblem",butitsabuseinthefinancialindustry.[43] Seealso[edit] Binomialoptionsmodel,adiscretenumericalmethodforcalculatingoptionprices Blackmodel,avariantoftheBlack–Scholesoptionpricingmodel BlackShoals,afinancialartpiece Brownianmodeloffinancialmarkets Financialmathematics(containsalistofrelatedarticles) Fuzzypay-offmethodforrealoptionvaluation Heatequation,towhichtheBlack–ScholesPDEcanbetransformed Jumpdiffusion MonteCarlooptionmodel,usingsimulationinthevaluationofoptionswithcomplicatedfeatures Realoptionsanalysis Stochasticvolatility Notes[edit] ^Althoughtheoriginalmodelassumednodividends,trivialextensionstothemodelcanaccommodateacontinuousdividendyieldfactor. References[edit] ^"Scholesonmerriam-webster.com".RetrievedMarch26,2012. ^abBodie,Zvi;AlexKane;AlanJ.Marcus(2008).Investments(7th ed.).NewYork:McGraw-Hill/Irwin.ISBN 978-0-07-326967-2. ^Taleb,1997.pp.91and110–111. ^Mandelbrot&Hudson,2006.pp.9–10. ^Mandelbrot&Hudson,2006.p.74 ^Mandelbrot&Hudson,2006.pp.72–75. ^Derman,2004.pp.143–147. ^Thorp,2017.pp.183–189. ^MacKenzie,Donald(2006).AnEngine,NotaCamera:HowFinancialModelsShapeMarkets.Cambridge,MA:MITPress.ISBN 0-262-13460-8. ^"TheSverigesRiksbankPrizeinEconomicSciencesinMemoryofAlfredNobel1997". ^"NobelPrizeFoundation,1997"(Pressrelease).October14,1997.RetrievedMarch26,2012. ^Black,Fischer;Scholes,Myron(1973)."ThePricingofOptionsandCorporateLiabilities".JournalofPoliticalEconomy.81(3):637–654.doi:10.1086/260062.S2CID 154552078. ^Merton,Robert(1973)."TheoryofRationalOptionPricing".BellJournalofEconomicsandManagementScience.4(1):141–183.doi:10.2307/3003143.hdl:10338.dmlcz/135817.JSTOR 3003143. ^abcdeNielsen,LarsTyge(1993)."UnderstandingN(d1)andN(d2):Risk-AdjustedProbabilitiesintheBlack–ScholesModel"(PDF).RevueFinance(JournaloftheFrenchFinanceAssociation).14(1):95–106.RetrievedDec8,2012,earliercirculatedasINSEADWorkingPaper92/71/FIN(1992);abstractandlinktoarticle,publishedarticle.CS1maint:postscript(link) ^DonChance(June3,2011)."DerivationandInterpretationoftheBlack–ScholesModel"(PDF).RetrievedMarch27,2012. ^Hull,JohnC.(2008).Options,FuturesandOtherDerivatives(7th ed.).PrenticeHall.ISBN 978-0-13-505283-9. ^Althoughwithsignificantalgebra;see,forexample,Hong-YiChen,Cheng-FewLeeandWeikangShih(2010). DerivationsandApplicationsofGreekLetters:ReviewandIntegration,HandbookofQuantitativeFinanceandRiskManagement,III:491–503. ^"ExtendingtheBlackScholesformula".finance.bi.no.October22,2003.RetrievedJuly21,2017. ^AndréJaun."TheBlack–ScholesequationforAmericanoptions".RetrievedMay5,2012. ^BerntØdegaard(2003)."ExtendingtheBlackScholesformula".RetrievedMay5,2012. ^DonChance(2008)."Closed-FormAmericanCallOptionPricing:Roll-Geske-Whaley"(PDF).RetrievedMay16,2012. ^GiovanniBarone-Adesi&RobertEWhaley(June1987)."EfficientanalyticapproximationofAmericanoptionvalues".JournalofFinance.42(2):301–20.doi:10.2307/2328254.JSTOR 2328254. ^BerntØdegaard(2003)."AquadraticapproximationtoAmericanpricesduetoBarone-AdesiandWhaley".RetrievedJune25,2012. ^DonChance(2008)."ApproximationOfAmericanOptionValues:Barone-Adesi-Whaley"(PDF).RetrievedJune25,2012. ^PetterBjerksundandGunnarStensland,2002.ClosedFormValuationofAmericanOptions ^Americanoptions ^Crack,TimothyFalcon(2015).HeardontheStreet:QuantitativeQuestionsfromWallStreetJobInterviews(16th ed.).TimothyCrack.pp. 159–162.ISBN 9780994118257. ^Hull,JohnC.(2005).Options,FuturesandOtherDerivatives.PrenticeHall.ISBN 0-13-149908-4. ^Breeden,D.T.,&Litzenberger,R.H.(1978).Pricesofstate-contingentclaimsimplicitinoptionprices.Journalofbusiness,621-651. ^Gatheral,J.(2006).Thevolatilitysurface:apractitioner'sguide(Vol.357).JohnWiley&Sons. ^Yalincak,Hakan(2012)."CriticismoftheBlack–ScholesModel:ButWhyIsItStillUsed?(TheAnswerisSimplerthantheFormula".SSRN 2115141.Citejournalrequires|journal=(help) ^abPaulWilmott(2008):IndefenceofBlackScholesandMertonArchived2008-07-24attheWaybackMachine,DynamichedgingandfurtherdefenceofBlack–Scholes[permanentdeadlink] ^RiccardoRebonato(1999).Volatilityandcorrelationinthepricingofequity,FXandinterest-rateoptions.Wiley.ISBN 0-471-89998-4. ^Kalotay,Andrew(November1995)."TheProblemwithBlack,Scholesetal"(PDF).DerivativesStrategy. ^EspenGaarderHaugandNassimNicholasTaleb(2011).OptionTradersUse(very)SophisticatedHeuristics,NevertheBlack–Scholes–MertonFormula.JournalofEconomicBehaviorandOrganization,Vol.77,No.2,2011 ^Boness,AJames,1964,Elementsofatheoryofstock-optionvalue,JournalofPoliticalEconomy,72,163–175. ^APerspectiveonQuantitativeFinance:ModelsforBeatingtheMarket,QuantitativeFinanceReview,2003.AlsoseeOptionTheoryPart1byEdwardThorpe ^EmanuelDermanandNassimTaleb(2005).TheillusionsofdynamicreplicationArchived2008-07-03attheWaybackMachine,QuantitativeFinance,Vol.5,No.4,August2005,323–326 ^Seealso:DorianaRuffinnoandJonathanTreussard(2006).DermanandTaleb'sTheIllusionsofDynamicReplication:AComment,WP2006-019,BostonUniversity-DepartmentofEconomics. ^[1] ^InPursuitoftheUnknown:17EquationsThatChangedtheWorld.NewYork:BasicBooks.13March2012.ISBN 978-1-84668-531-6. ^Nahin,PaulJ.(2012)."InPursuitoftheUnknown:17EquationsThatChangedtheWorld".PhysicsToday.Review.65(9):52–53.Bibcode:2012PhT....65i..52N.doi:10.1063/PT.3.1720.ISSN 0031-9228. ^abStewart,Ian(February12,2012)."Themathematicalequationthatcausedthebankstocrash".TheGuardian.TheObserver.ISSN 0029-7712.RetrievedApril29,2020. Primaryreferences[edit] Black,Fischer;MyronScholes(1973)."ThePricingofOptionsandCorporateLiabilities".JournalofPoliticalEconomy.81(3):637–654.doi:10.1086/260062.S2CID 154552078.[2](BlackandScholes'originalpaper.) Merton,RobertC.(1973)."TheoryofRationalOptionPricing".BellJournalofEconomicsandManagementScience.TheRANDCorporation.4(1):141–183.doi:10.2307/3003143.hdl:10338.dmlcz/135817.JSTOR 3003143.[3] Hull,JohnC.(1997).Options,Futures,andOtherDerivatives.PrenticeHall.ISBN 0-13-601589-1. Historicalandsociologicalaspects[edit] Bernstein,Peter(1992).CapitalIdeas:TheImprobableOriginsofModernWallStreet.TheFreePress.ISBN 0-02-903012-9. Derman,Emanuel."MyLifeasaQuant"JohnWiley&Sons,Inc.2004.ISBN 0471394203 MacKenzie,Donald(2003)."AnEquationanditsWorlds:Bricolage,Exemplars,DisunityandPerformativityinFinancialEconomics"(PDF).SocialStudiesofScience.33(6):831–868.doi:10.1177/0306312703336002.hdl:20.500.11820/835ab5da-2504-4152-ae5b-139da39595b8.S2CID 15524084.[4] MacKenzie,Donald;YuvalMillo(2003)."ConstructingaMarket,PerformingTheory:TheHistoricalSociologyofaFinancialDerivativesExchange".AmericanJournalofSociology.109(1):107–145.CiteSeerX 10.1.1.461.4099.doi:10.1086/374404.S2CID 145805302.[5] MacKenzie,Donald(2006).AnEngine,notaCamera:HowFinancialModelsShapeMarkets.MITPress.ISBN 0-262-13460-8. Mandelbrot&Hudson,"The(Mis)BehaviorofMarkets"BasicBooks,2006.ISBN 9780465043552 Szpiro,GeorgeG.,PricingtheFuture:Finance,Physics,andthe300-YearJourneytotheBlack–ScholesEquation;AStoryofGeniusandDiscovery(NewYork:Basic,2011)298pp. Taleb,Nassim."DynamicHedging"JohnWiley&Sons,Inc.1997.ISBN 0471152803 Thorp,Ed."AManforallMarkets"RandomHouse,2017.ISBN 9781400067961 Furtherreading[edit] Haug,E.G(2007)."OptionPricingandHedgingfromTheorytoPractice".Derivatives:ModelsonModels.Wiley.ISBN 978-0-470-01322-9.ThebookgivesaseriesofhistoricalreferencessupportingthetheorythatoptiontradersusemuchmorerobusthedgingandpricingprinciplesthantheBlack,ScholesandMertonmodel. Triana,Pablo(2009).LecturingBirdsonFlying:CanMathematicalTheoriesDestroytheFinancialMarkets?.Wiley.ISBN 978-0-470-40675-5.ThebooktakesacriticallookattheBlack,ScholesandMertonmodel. Externallinks[edit] Discussionofthemodel[edit] AjayShah.Black,MertonandScholes:Theirworkanditsconsequences.EconomicandPoliticalWeekly,XXXII(52):3337–3342,December1997 ThemathematicalequationthatcausedthebankstocrashbyIanStewartinTheObserver,February12,2012 WhenYouCannotHedgeContinuously:TheCorrectionstoBlack–Scholes,EmanuelDerman TheSkinnyOnOptionsTastyTradeShow(archives) Derivationandsolution[edit] DerivationoftheBlack–ScholesEquationforOptionValue,Prof.ThayerWatkins SolutionoftheBlack–ScholesEquationUsingtheGreen'sFunction,Prof.DennisSilverman SolutionviariskneutralpricingorviathePDEapproachusingFouriertransforms(includesdiscussionofotheroptiontypes),SimonLeger Step-by-stepsolutionoftheBlack–ScholesPDE,planetmath.org. TheBlack–ScholesEquationExpositoryarticlebymathematicianTerenceTao. Computerimplementations[edit] Black–ScholesinMultipleLanguages Black–ScholesinJava-movingtolinkbelow- Black–ScholesinJava ChicagoOptionPricingModel(GraphingVersion) Black–Scholes–MertonImpliedVolatilitySurfaceModel(Java) OnlineBlack–ScholesCalculator Historical[edit] TrillionDollarBet—CompanionWebsitetoaNovaepisodeoriginallybroadcastonFebruary8,2000."ThefilmtellsthefascinatingstoryoftheinventionoftheBlack–ScholesFormula,amathematicalHolyGrailthatforeveralteredtheworldoffinanceandearneditscreatorsthe1997NobelPrizeinEconomics." 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