Black-Scholes-Merton | Brilliant Math & Science Wiki

The Black-Scholes-Merton model, sometimes just called the Black-Scholes model, is a mathematical model of financial derivative markets from which the ... Brilliant Today Courses Signup Login Thisholidayseason,sparkalifelongloveoflearning. GiftBrilliantPremium Excelinmathandscience. LoginwithFacebook LoginwithGoogle Loginwithemail JoinusingFacebook JoinusingGoogle Joinusingemail Forgotpassword? Newuser? Signup Existinguser? Login SignupwithFacebook or Signupmanually Alreadyhaveanaccount? Loginhere. ChristopherWilliams, AFormerBrilliantMember, PatrickCorn, and 2others JiminKhim CalvinLin contributed TheBlack-Scholes-Mertonmodel,sometimesjustcalledtheBlack-Scholesmodel,isamathematicalmodeloffinancialderivativemarketsfromwhichtheBlack-Scholesformulacanbederived.Thisformulaestimatesthepricesofcallandputoptions.Originally,itpricedEuropeanoptionsandwasthefirstwidelyadoptedmathematicalformulaforpricingoptions.Somecreditthismodelforthesignificantincreaseinoptionstrading,andnameitasignificantinfluenceinmodernfinancialpricing.Priortotheinventionofthisformulaandmodel,optionstradersdidn'talluseaconsistentmathematicalwaytovalueoptions,andempiricalanalysishasshownthatpriceestimatesproducedbythisformulaareclosetoobservedprices. Intheirinitialformulationofthemodel,FischerBlackandMyronScholes(theeconomistswhooriginallyformulatedthemodel)cameupwithapartialdifferentialequationknownastheBlack-Scholesequation[1],andlaterRobertMertonpublishedamathematicalunderstandingoftheirmodel,usingstochasticcalculus[2]thathelpedtoformulatewhatbecameknownastheBlack-Scholes-Mertonformula.BothMyronScholesandRobertMertonsplitthe1997NobelPrizeinEconomists,listingFischerBlackasacontributor,thoughhewasineligiblefortheprizeashehadpassedawaybeforeitwasawarded. Roughly,theirmodeldeterminesthepriceofanoptionbycalculatingthereturnaninvestorgetslesstheamountthatinvestorhastopay,usinglog-normaldistributionprobabilitiestoaccountforvolatilityintheunderlyingasset.Thelog-normaldistributionofreturnsusedinthemodelisbasedontheoriesofBrownianmotion,withassetpricesexhibitingsimilarbehaviortotheorganicmovementinBrownianmotion. Theformulahelpedtolegitimizeoptionstrading,makingitseemlesslikegamblingandmorelikescience.Today,theBlack-Scholes-Mertonformulaiswidelyused,thoughinindividuallymodifiedways,bytradersandinvestors,asitisthefundamentalstrategyofhedgingtobestcontrol,or"eliminate",risksassociatedwithvolatilityintheassetsthatunderlietheoption. Contents TheBlack-Scholes-MertonFormula OnVolatility High-levelExplanationoftheBlack-Scholes-MertonFormula Example+Problem Hedgingto"Eliminate"Risk CriticismsoftheBlack-Scholes-MertonModel Discussion References Again,theBlack-Scholes-MertonformulaisanestimateofthepricesofEuropeancallandputoptions,withthecoredifferencebetweenAmericanandEuropeanoptionsbeingthatEuropeanoptionscanonlybeexercisedontheironeexercisedateversusAmericancalloptionsthatcanbeexercisedanytimeuptothatexpirationdate.It'salsousedonlytodeterminepricesofnon-dividendpayingassets. TheBlack–Scholes-MertonformulaofvalueforaEuropeancalloptionis(note:theformulaforaEuropeanputoptionissimilar) C(S0,t)=S0N(d1)−Ke−r(T−t)N(d2),C(S_0,t)=S_0N(d_1)-Ke^{-r(T-t)}N(d_2),C(S0​,t)=S0​N(d1​)−Ke−r(T−t)N(d2​), where S0S_0S0​isthestockprice; C(S0,t)C(S_0,t)C(S0​,t)isthepriceofthecalloptionasaformulationofthestockpriceandtime; KKKistheexerciseprice; (T−t)(T-t)(T−t)isthetimetomaturity,i.e.theexercisedateTTT,lesstheamountoftimebetweennowtttandthen.Generally,thisisrepresentedinyearswithonemonthequaling112\frac{1}{12}121​or0.083‾;0.08\overline{3};0.083; N(d1)N(d_1)N(d1​)andN(d2)N(d_2)N(d2​)arecumulativedistributionfunctionsforastandardnormaldistributionwiththefollowingformulation: d1=ln⁡S0K+(r+σ22)(T−t)σT−td2=d1−σ(T−t)=ln⁡S0K+(r−σ22)(T−t)σT−t,\begin{aligned}d_1&=\frac{\ln{\frac{S_0}{K}}+\big(r+\frac{\sigma^2}{2}\big)(T-t)}{\sigma\sqrt{T-t}}\\ d_2&=d_1-\sigma\sqrt{(T-t)}\\&=\frac{\ln{\frac{S_0}{K}}+\big(r-\frac{\sigma^2}{2}\big)(T-t)}{\sigma\sqrt{T-t}},\end{aligned}d1​d2​​=σT−t​lnKS0​​+(r+2σ2​)(T−t)​=d1​−σ(T−t)​=σT−t​lnKS0​​+(r−2σ2​)(T−t)​,​ where σ\sigmaσrepresentstheunderlyingvolatility(astandarddeviationoflogreturns); rrristherisk-freeinterestrate,i.e.therateofreturnaninvestorcouldgetonaninvestmentassumedtoberisk-free(likeaT-bill). PriceofanOilandaCowETFoverthreeyears Volatility,inthecaseoffinancialassets,isthemeasureofhowmuchandhowquicklytheasset'spricechanges.It'sameasureofuncertainty.Iftraderswerecertainthatanassetwasworthacertainamount,thenthey'dbuyatthatpriceandsellbelowit.They'dsitonit.Buthighlyuncertainassetsgettradedatawiderrangeofprices.Impliedvolatility,whatoptionsuse,isthevalueofthevolatilityoftheunderlyingasset. PriortotheBlack-Scholes-Mertonformula,investorshadtheirownwaysofestimatingthepriceofoptions.Thesemethodsvaried,butgenerallytheyincorporatedsomemeasureofimpliedvolatility.Stockswithmorevolatilityhadahigherchanceofhavingaveryhighvalueinthefuture,oraverylowvalue.Intheabovegraph,anOILETFandanETFforLiveCattle(COW)aregraphed,withtheOilETFbeingamuchmorevolatileasset(spikingupanddownmore).Thepricenotonlydecreasesmoredrastically(largerslopeinthetrendline),butonadailybasisthestockfluctuatesmoresignificantly(thepricefluctuatesfromthetrendlinemoredrastically). Becauseofthewayoptionswork,thebuyerofacallonlymakesmoneyifanassetisabovethestrikeprice.Ifit'sbelowthatprice,theydon'tcarehowmuchbelowthestrikeitis,theyhavespentthesameamount.Buttheydocarehowmuchabovethestrikepriceitis.Assuch,highlyvolatileassets(optionswithhigherimpliedvolatility)aremorelikelytomakeinvestorsmoremoney,andaremorevaluable. Technically,andinthecaseoftheBlack-Scholes-Mertonmodel,impliedvolatilityistheannualizedstandarddeviationofthereturnontheasset,andisexpressedasadecimalpercentage.Thiswillbeexplainedmorebelow.ButintheB-S-Mformula,σ\sigmaσisbothameasureofimpliedvolatilityandthestandarddeviation.Thisisbecause,inameasureofpossiblereturnsforanasset,highlyvolatileassetswillhaveawiderstandarddeviationthanlessvolatileones.ThegraphbelowshowstheperiodicdailyreturnsforthepreviousCowandOilETF,essentiallyacountofhowmanydaysthepricechanged+1%+1\%+1%,or0%0\%0%,or−2%-2\%−2%,etc.BecausetheCowETFisalessvolatilestock,thegraphofitsnormaldistributionisnarrower,andthestandarddeviationislowerat~3.73.73.7;expressedasapercentagethat's13.8%13.8\%13.8%fromthemean,meaningthatthepriceoftheETFonanygivenday(becausethisisagraphofperiodicdailyreturns)ishighlyunlikelytobe±13.8%\pm13.8\%±13.8%morethanthemean.Onestandarddeviationinanormaldistributionis68.2%68.2\%68.2%,andthemeanwas$26.85\$26.85$26.85,sotheexpectedpricesfor68.2%68.2\%68.2%ofthedayswere$23.13≤p≤$30.56\$23.13\lep\le\$30.56$23.13≤p≤$30.56.Andindeed,comparedtothis,theOilETF'sgraphismuchwider;infactitgoesbeyondthecurrentxxx-axisofthisgraph(thereisawidergraphofthissameETFoverthissametimeperiodinthesectionbelow).Ithasastandarddeviationof 7.5~7.5 7.5,or55.4%55.4\%55.4%.TheOilETFhadameanof$13.58\$13.58$13.58,sofor68.2%68.2\%68.2%oftheday'spriceswere$6.03≤p≤$21.01\$6.03\lep\le\$21.01$6.03≤p≤$21.01,amuchwiderrangeofprices(bothabsolutelyandrelatively). Overall:Intuitively,androughly,theBlack-Scholes-MertonformulasubtractsKe−r(T−t)N(d2)Ke^{-r(T-t)}N(d_2)Ke−r(T−t)N(d2​),theexercisepricediscountedbacktopresentvaluetimestheprobabilitythattheoptionisabovethestrikepriceatmaturity,fromS0N(d1)S_0N(d_1)S0​N(d1​),thestockpricetodaytimesaprobabilitythatis000ifthestockisbelowthestrikepricebutissomeprobabilityrepresentingthestock'svalueifit'sabovethestrikeprice.Roughly,it'saninvestor'sreturn,minusthecostoftheoption. DiscountingtoPresentValue:Theer(T−t)e^{r(T-t)}er(T−t)portionoftheformulationissimplyacalculationofthepresentvalueofthatstrikeprice.Itcompoundstherisk-freeinterestrateovertheperiodbetweennow(whenthecalculationisdone)andthefutureexpirationdate.Thisisdonebecausethepriceofthisoptionshouldreflectthatalternativerisk-freechoiceaninvestorhas.Ifaninvestorcouldputsomemoneyinarisk-freeT-Billandgeta2%2\%2%returnoveroneyear,thenanoptionneedstogenerateanadditionalreturnaboveandbeyondthat2%2\%2%tojustifytheincreasedriskassociatedwithit. TheperiodicdailyreturnsforanOilETFforeverydayofthethreeyearsbetweenNov1,2013andNov1,2016.Roughly,thisdistributionshowstheamountofeachday'sgainorlossandthenumberoftimesthatgain/losshappened.Forinstance,therewere98dayswith0%gainorlossinthis755tradingdayperiod. Probability:Thoseprobabilityweightings,N(d1)N(d_1)N(d1​)andN(d2)N(d_2)N(d2​),comefromanormalprobabilitydistributioncurve.Ifaninvestorgraphedtheperiodicdailyreturns(thereturnsforthisoptioneachday)theresultinggraphwouldbeanormaldistribution,abell-shapedcurve,liketheonefortheOilETFtotheright.Justasthehistoricalpriceswerenormallydistributed,theB-S-Mmodelassumesthatfuturepriceswillbenormallydistributed.ThereforeN(d1)N(d_1)N(d1​)is,roughly,lookingforN(z-score)N(z\text{-score})N(z-score),theareaunderthebellcurveuptosomezzz-score,ortheprobabilitythatthefuturepricewillbeabovethestrikepriceontheexpirationdate.Thestandardnotationforzzz-scoreis z-score=x−μσ.z\text{-score}=\frac{x-\mu}{\sigma}.z-score=σx−μ​. TheN(d1)N(d_1)N(d1​)andN(d2)N(d_2)N(d2​)functionsaresimplycalculationsofareaonthecurve.Forinstance,acalculationofN(d2)N(d_2)N(d2​)forthisOilETFisrepresentedintheimagetotheright.Ifthecurveisthenormaldistributionofallprobabilitiesfortheoption,thenN(d2)N(d_2)N(d2​)isthepercentageofprobabilitiesthattheoptionwillexpireinthemoney. Giventhecomplexityofthemodel,it'salwaysgoodtoseeitinaction: Smartinvestorscalculatethepriceofanoptionforthemselvesbeforetheybuy.IfyouhavethechancetobuyaEuropeancalloptionwiththefollowingparameters,whatcostshouldyoupaylessthantomakeitworthit? stockprice:$50 strikeprice:$45 timetoexpiration:80days risk-freeinterestrate:2% impliedvolatility:30% Inotherwords,usingtheB-S-Mformula,whatshouldthecostofthiscalloptionbe? Whiletheformulaisinvolved,thisisessentiallyamatterofplugginginthegivenvariables: d1=ln⁡S0K+(r+σ22)(T−t)σT−t=ln⁡5045+(0.02+.322)(80365)0.380365=0.105+0.0140.140≈0.851d2=d1−σT−t=0.851−0.380365≈0.711.\begin{aligned} d_1&=\frac{\ln{\frac{S_0}{K}}+\big(r+\frac{\sigma^2}{2}\big)(T-t)}{\sigma\sqrt{T-t}}\\\\ &=\frac{\ln{\frac{50}{45}}+\big(0.02+\frac{.3^2}{2}\big)\big(\frac{80}{365}\big)}{0.3\sqrt{\frac{80}{365}}}\\\\ &=\frac{0.105+0.014}{0.140}\approx0.851\\\\\\ d_2&=d_1-\sigma\sqrt{T-t}\\ &=0.851-0.3\sqrt{\frac{80}{365}}\approx0.711. \end{aligned}d1​d2​​=σT−t​lnKS0​​+(r+2σ2​)(T−t)​=0.336580​​ln4550​+(0.02+2.32​)(36580​)​=0.1400.105+0.014​≈0.851=d1​−σT−t​=0.851−0.336580​​≈0.711.​ N(d1)N(d_1)N(d1​)andN(d2)N(d_2)N(d2​)canbefoundbylookingatazzz-scoretable: N(d1)=0.8023N(d2)=0.7611⇒C(S0,t)=S0N(d1)−Ke−r(T−t)N(d2)=($50×0.8023)−($45×e(−0.02×80365)×0.7611)=$40.12−$34.10=$6.02. □\begin{aligned} N(d_1)&=0.8023\\ N(d_2)&=0.7611\\\\ \RightarrowC(S_0,t)&=S_0N(d_1)-Ke^{-r(T-t)}N(d_2)\\ &=(\$50\times0.8023)-\left(\$45\timese^{\big(-0.02\times\frac{80}{365}\big)}\times0.7611\right)\\ &=\$40.12-\$34.10\\ &=\$6.02.\_\square \end{aligned}N(d1​)N(d2​)⇒C(S0​,t)​=0.8023=0.7611=S0​N(d1​)−Ke−r(T−t)N(d2​)=($50×0.8023)−($45×e(−0.02×36580​)×0.7611)=$40.12−$34.10=$6.02. □​​ Note:Inthecaseofthisstock,thereisaprobabilityof≈80%\approx80\%≈80%thatrepresentstheexpectedvalueatexpiration,whichismultipliedby$50\$50$50toyielda$40.12\$40.12$40.12return.Thestrikepriceis$45\$45$45anditspresentvalueis$44.80,\$44.80,$44.80,whichismultipliedby≈76%,\approx76\%,≈76%,probabilityofthecalloptionexpiringinthemoney,toyield$34.10\$34.10$34.10. 1.0% 10.0% 15.7% 16.5% 25.2% 25.4% 50.6% Supposeyou'reaninvestorandarecuriouswhatthemarketthinkstheimpliedvolatilityoftheS&P500istoday.Youknowafewthings: YoucanassumethateveryotherinvestorisusingBlack-Scholes-Mertonformulaforpricing. YoulookacommonETFoftheS&P500,theSPYspider. Todayit'spricedat$216. AEuropeancalloptionhasastrikepriceof$210. Toexpire,thereare30daysleftfromtoday. Therisk-freeinterestrateis1.8%. ThemarketispricingthisEuropeancalloptionat$7.93. Whatistheimpliedvolatility? (Allanswersaretruncated.) Note:UsingExcel's"GoalSeek"maybehelpful. Onceanassetispriced,thekeyideaistohedgetheoptionbybuyingandsellingtheunderlyingassetinjusttherightwaysoasto"eliminaterisk."Thisisreferredtoasdeltahedgingordynamichedging.TheideaistomaintainazerooptionGreeks—delta,wheredeltaisthesensitivityofanoptiontochangesinthepriceoftheunderlyingassets.Thisisafairlycomplexformofhedging,andisprincipallyperformedbylargeinvestmentinstitutions(investmentbanks,hedgefunds,privateequityfunds,etc.).Theformalcalculationfordeltais Δ=∂V∂S.\Delta=\frac{\partialV}{\partialS}.Δ=∂S∂V​. Thatisthefirstderivativeofthevalueoftheoptionoverthefirstderivativeofthevalueoftheunderlyingasset.Assuch,thebasicstrategyofdeltahedgingistobuyorsellsomeoftheunderlyingasset(thedenominator)inresponsestochangesinthevalueofthatasset;itistokeep∂S{\partialS}∂Sstatic,eventhoughthevalueofthatassetchangeregularly. Oneimportantnoteisthattheriskeliminatedhereisnottheriskthattheunderlyingassetwillgodowninvalue;thisisnottopreventnormalnegativereturnsfromassetssimplynotperforming,buttoeliminatethemoresharpshiftsinpricenotcorrelatedtochangesinunderlyingvalue—thetailendsofthatperiodicdailyreturngraphabovewhereinasingledaythepriceofanassetcanchangesignificantly,andthencorrectbacktotheoriginalpriceinfollowingdays.Also,riskisneverreally"eliminated."Thatisthegoal,buttherearealwaysrisks,likedefaultrisk,thatarehardertocontrolfor. NassimNicholasTaleb,famousforhis2007bestsellingbook"BlackSwan"whichdiscussedunpredictableeventsinfinancialmarkets,alongwithEspenGaarderHaughascriticizedtheBlack-Scholes-Mertonmodel,sayingthatitis"fragiletojumpsandtailevents"andcanonlyhandle"mildrandomness."[3]Thisisoneofafewknownchallengestothemodel: Fragilityto"tail-risk"orotherextremerandomness:Ingeneral,returnsdonotabsolutelyfollowanormaldistribution.Theppp-valueontheAnderson-Darlingnormalitytestis0.000whenappliedtoS&Preturns,showingthatmarketreturnsareleptokurtic(havinggreaterkurtosis,ormoreconcentratedaboutthemeanwithfattails).[4] ThestructureofB-S-Mdoesn'treflectpresentrealities:TheB-S-MmodelassumesamarketusingEuropeancalloptionswhenmostoptionstradedtodayareAmericancalloptionsthatcanbesoldatanypoint.Italsodoesnotallowfordividends,somethingthatiscommonlyfoundinoptions. Assumptionofarisk-freeinterestrate:Atheoreticalcalculationofrisk-freeratesishardtocomeupwithand,inpractice,investorsuseproxieslikethelong-termyieldontheUSTreasurycouponbonds(generally10-yearbonds).However,thisassumesthatUSTreasurybondsare"risk-free"whenamoreaccuratestatementwouldbethatthey'rewhatthemarketassumestheleastriskyinvestmentvehicles. Assumptionofcostlesstrading:Tradinggenerallycomeswithexchangefees,thecoststobuyorsellstocksandoptions,andthecostoftime;thetimeittakesfortheordertogothroughmayresultinchangestothepriceonthemarket.Thesecostscanbemanaged,butarenotincludedinthemodel. Gaprisk:Alsothemodelassumesthattradingoccurscontinuously,unlikereality,wheremarketsshutdownforthenightandthencanreopenatsignificantlydifferentpricestoreflectnewinformation. Empirically,significantpricingdiscrepanciesbetweenB-S-Mandrealityhaveoccurredmoreoftenthanifreturnswerelog-normal.ButtheB-S-Mmodelcontinuestobeused.Itissimple,easytodetermine,andcanbeadjustedforvariousinadequacies. AVolatilitySmile[5] OneofthemorecommoncriticismsoftheB-S-Mmodelistheexistenceofavolatilitysmile.TheBlack-Scholes-Mertonpricingmodelsuggestsaconstantvolatilityandlog-normaldistributionsofreturns,where,inreality,impliedvolatilityvarieswidely.Optionswhosestrikepricearesaidtobe"deep-in-the-money"or"out-of-the-money,"i.e.whosestrikepriceisfurtherawayfromtheassumedunderlyingassetprice,commandhigherpricesthanaflatvolatilitywouldsuggest—theirimpliedvolatilityishigher. Thenotationisnotstandardmathematicalnotationbutisthestandardformsusedinthefinanceindustry. Whatiscalledanormaldistributionisnotanormaldistribution;rather,itisthecumulativedistributionfunctionofalog-normaldistribution.Theuseofanunderlyingnormaldistributionwithameanof0andastandarddeviationof1isassumedandseldommentioned. Theuseofthelog-normaldistributionisbecausethecompoundinterest,whichisapowerlaw,isbeingmodeled.Takingthelogsofthegrowthfactorsmakesthegrowthfactorsnearlylinearandthedistributionnearlynormal.Thevaluesofmumumuandσ\sigmaσaretheexpectedgrowthfactor(interestrate)andtheexpectedstandarddeviation(volatility)foronetimeperiod.Therefore,valuescloseto0areexpected. Continuousfunctionsareusedtomodeldiscretefunctionstosimplifythecomputationswithoutwarning,e.g.dividendsandinterestcomputedcontinuallyandnotperiodically.Thisfactisnotmentionedinthediscussion.Mathematiciansdothisalso,buttheygenerallymentionthepractice. Whatisbeingmodeledisarandomone-dimensionalwalkormartingale.Sinceabinomialdistributionmodelsanormaldistributionoveralargenumberoftrials,e.g.thechangesinpricesoverayear'stime,thismodelingofthenormaldistributionisareasonableapproximation. Theuseofthelogarithmfunctionrequiresthattheargumentbeapositiverealtoavoidinfinitiesandcomplexnumbers: Log Normal Distribution[μ,σ]=Transformed Distribution[exp⁡(x),x≈Normal Distribution[μ,σ]]\text{LogNormalDistribution}\big[\mu,\sigma\big]= \text{TransformedDistribution}\big[\exp(x),x\approx\text{NormalDistribution}[\mu,\sigma]\big]Log Normal Distribution[μ,σ]=Transformed Distribution[exp(x),x≈Normal Distribution[μ,σ]] PDF[NormalDistribution[μ,σ]]⇒x→e−(x−μ)22σ22πσ\text{PDF}[\text{NormalDistribution}[\mu,\sigma]]\Rightarrowx\to\frac{e^{-\frac{(x-\mu)^2}{2\sigma^2}}}{\sqrt{2\pi}\sigma}PDF[NormalDistribution[μ,σ]]⇒x→2π​σe−2σ2(x−μ)2​​ PDF[LogNormalDistribution[μ,σ]]⇒\text{PDF}[\text{LogNormalDistribution}[\mu,\sigma]]\RightarrowPDF[LogNormalDistribution[μ,σ]]⇒ x→x\tox→ e−(log⁡(x)−μ)22σ2x2πσx>00True \begin{array}{cc} \frac{e^{-\frac{(\log(x)-\mu)^2}{2\sigma^2}}}{x\sqrt{2\pi}\sigma}&x>0\\ 0&\text{True}\\ \end{array} x2π​σe−2σ2(log(x)−μ)2​​0​x>0True​ Mean[LogNormalDistribution[μt,σt]]=Expectation[x,x∼∼LogNormalDistribution[μt,σt]]=eσ2t22+μt\text{Mean}[\text{LogNormalDistribution}[\mut,\sigmat]]= \text{Expectation}[x,x\overset{\sim}{\sim}\text{LogNormalDistribution}[\mut,\sigmat]]= e^{\frac{\sigma^2t^2}{2}+\mut}Mean[LogNormalDistribution[μt,σt]]=Expectation[x,x∼∼LogNormalDistribution[μt,σt]]=e2σ2t2​+μt UsingmaterialfromWolframMathematica12Expectationdocumentation: expectedPrice[currentPrice,μ,σ,t]=currentPrice×Mean[LognormalDistribution[μ t,σ t]]⇒currentPrice ⁣×eμt+σ22 t\text{expectedPrice}[\text{currentPrice},\mu,\sigma,t]=\\ \text{currentPrice}\times\text{Mean}[\text{LognormalDistribution}[\mu\,t,\sigma\,t]]\Rightarrow\\ \text{currentPrice}\!\times\mathbb{e}^{\mut+\frac{{\sigma}^2}{2}\,t}expectedPrice[currentPrice,μ,σ,t]=currentPrice×Mean[LognormalDistribution[μt,σt]]⇒currentPrice×eμt+2σ2​t "Assuminganinvestorcaninvestmoneyinastockwithdividendyieldqqqforayearatacontinuouslycompoundedyearlyraterrr risk-free,therisk-neutralpricingconditionrequires:" WiththecurrentPrice\text{currentPrice}currentPricefactoredoutofbothsidesoftheequationandtheincreaseofvaluecausedbyrisk-freeinterestratelesstheeffectiveinterestratefromthedividendyield,assumingthatbothratesarecompoundedcontinuously: expectedPrice(currentPrice,μ,σ,t+1)=er−q expectedPrice(currentPrice,μ,σ,t)\text{expectedPrice}\left(\text{currentPrice},\mu,\sigma,t+1\right)=\mathbb{e}^{r-q}\,\text{expectedPrice}\left(\text{currentPrice},\mu,\sigma,t\right)expectedPrice(currentPrice,μ,σ,t+1)=er−qexpectedPrice(currentPrice,μ,σ,t) eμ (t+1)+12σ2(t+1)=e−q+r+μ t+σ2t2\mathbb{e}^{\mu\,(t+1)+\frac{1}{2}\sigma^2(t+1)}=\mathbb{e}^{-q+r+\mu\,t+\frac{\sigma^2t}{2}}eμ(t+1)+21​σ2(t+1)=e−q+r+μt+2σ2t​ Solvingforμ\muμoverallpositivetimegivesμ=12(−2q+2r−σ2)\mu=\frac{1}{2}\left(-2q+2r-\sigma^2\right)μ=21​(−2q+2r−σ2). "Consideracalloptiontobuythisstockayearfromnow,atafixedpriceK\mathcal{K}K.Thevalueofsuchanoptionis:" callOptVal(expectedPrice,K)=max⁡((expectedPrice−K,0)\text{callOptVal}(\text{expectedPrice},\mathcal{K})=\max((\text{expectedPrice}-\mathcal{K},0)callOptVal(expectedPrice,K)=max((expectedPrice−K,0) Thisisbecauseacalloptionisworthlessifanimmediateprofitcannotbemade. "[C]onsideraputoptiontosellthisstockayearfromnow,atafixedpriceK\mathcal{K}K.Thevalueofsuchanoptionis:" putOptVal(expectedPrice,K)=max⁡(K−expectedPrice,0)\text{putOptVal}(\text{expectedPrice},\mathcal{K})=\max(\mathcal{K}-\text{expectedPrice},0)putOptVal(expectedPrice,K)=max(K−expectedPrice,0) Thisisbecauseaputoptionisworthlessifanimmediateprofitcannotbemade. Intheformulasbelow,allparametersarepositivereal,μ\muμisascomputedaboveandthedistributionisasintheargumenttotheMeanfunctionabove: Thecalloptionpriceise−r tExpectation[callOptVal(f currentPrice,K]),f∼∼LognormalDistribution(μ,σ,t)]\mathbb{e}^{-r\,t}\text{Expectation}[\text{callOptVal}(f\,\text{currentPrice},\mathcal{K}]),f\overset{\sim}{\sim}\text{LognormalDistribution}(\mu,\sigma,t)]e−rtExpectation[callOptVal(fcurrentPrice,K]),f∼∼LognormalDistribution(μ,σ,t)]. Theputoptionpriceise−r tExpectation[putOptVal(f currentPrice,K]),f∼∼LognormalDistribution(μ,σ,t)]\mathbb{e}^{-r\,t}\text{Expectation}[\text{putOptVal}(f\,\text{currentPrice},\mathcal{K}]),f\overset{\sim}{\sim}\text{LognormalDistribution}(\mu,\sigma,t)]e−rtExpectation[putOptVal(fcurrentPrice,K]),f∼∼LognormalDistribution(μ,σ,t)]. BlackScholesCallOptionPrice(currentPrice,K,r,q,σ,t)=12e−rt(currentPrice×et(r−q)(erf(−2log⁡(K)+t(−2q+2r+σ2)+2log⁡(currentPrice)22σt)+1)−K erfc(2log⁡(K)+t(2q−2r+σ2)−2log⁡(currentPrice)22σt))\text{BlackScholesCallOptionPrice}(\text{currentPrice},\mathcal{K},r,q,\sigma,t)=\\ \frac{1}{2}\mathbb{e}^{-rt}\left(\text{currentPrice}\times\mathbb{e}^{t(r-q)}\left(\text{erf}\left(\frac{-2\log(\mathcal{K})+t\left(-2q+2r+\sigma^2\right)+2\log(\text{currentPrice})}{2\sqrt{2}\sigma\sqrt{t}}\right)+1\right)-\\ \mathcal{K}\,\text{erfc}\left(\frac{2\log(\mathcal{K})+t\left(2q-2r+\sigma^2\right)-2\log(\text{currentPrice})}{2\sqrt{2}\sigma\sqrt{t}}\right)\right)BlackScholesCallOptionPrice(currentPrice,K,r,q,σ,t)=21​e−rt(currentPrice×et(r−q)(erf(22​σt​−2log(K)+t(−2q+2r+σ2)+2log(currentPrice)​)+1)−Kerfc(22​σt​2log(K)+t(2q−2r+σ2)−2log(currentPrice)​)) BlackScholesCallOptionPrice(currentPrice,K,r,q,σ,t)=12e−rt(K erf(t(2q−2r+σ2)+2log⁡(KcurrentPrice)22σt)−currentPrice×et(r−q)erfc(−2log⁡(K)+t(−2q+2r+σ2)+2log⁡(currentPrice)22σt)+K)\text{BlackScholesCallOptionPrice}(\text{currentPrice},\mathcal{K},r,q,\sigma,t)=\\ \frac{1}{2}\mathbb{e}^{-rt}\left(\mathcal{K}\,\text{erf}\left(\frac{t\left(2q-2r+\sigma^2\right)+2\log\left(\frac{\mathcal{K}}{\text{currentPrice}}\right)}{2\sqrt{2}\sigma\sqrt{t}}\right)-\\ \text{currentPrice}\times\mathbb{e}^{t(r-q)}\text{erfc}\left(\frac{-2\log(\mathcal{K})+t\left(-2q+2r+\sigma^2\right)+2\log(\text{currentPrice})}{2\sqrt{2}\sigma\sqrt{t}}\right)+\mathcal{K}\right)BlackScholesCallOptionPrice(currentPrice,K,r,q,σ,t)=21​e−rt(Kerf(22​σt​t(2q−2r+σ2)+2log(currentPriceK​)​)−currentPrice×et(r−q)erfc(22​σt​−2log(K)+t(−2q+2r+σ2)+2log(currentPrice)​)+K) Thefunctionerf(z)=2π∫0ze−t2 dt\text{erf}(z)=\frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}\,dterf(z)=π​2​∫0z​e−t2dtanderfc(z)=1−erf(z)\text{erfc}(z)=1-\text{erf}(z)erfc(z)=1−erf(z).Bothfunctionsassumeanormaldistributionwithameanof0andastandarddeviationof1.Thismeansthattheargumentsmustbenormalizedtoaameanof0andastandarddeviationof1whichhasbeendoneinthecallandoptionvalueformulaeabove. CDF[NormalDistribution[0,1],x]=12erfc(−x2)\text{CDF}[\text{NormalDistribution}[0,1],x]=\frac{1}{2}\text{erfc}\left(-\frac{x}{\sqrt{2}}\right)CDF[NormalDistribution[0,1],x]=21​erfc(−2​x​) erfc(−z)=1+erf(z)\text{erfc}(-z)=1+\text{erf}(z)erfc(−z)=1+erf(z) Black,F., & Scholes,M. ThePricingofOptionsandCorporateLiabilities. Retrieved November1,2016, from http://www.jstor.org/stable/1831029 Merton,R. TheoryofRationalOptionPricing. Retrieved November1,2016, from http://www.jstor.org/stable/3003143 Haug,E., & Taleb,N. WhyWeHaveNeverUsedtheBlack–Scholes–MertonOptionPricingFormula. Retrieved November9th2016, from http://polymer.bu.edu/hes/rp-haug08.pdf Hurvich,C. SOMEDRAWBACKSOFBLACK-SCHOLES. Retrieved November9th2016, from http://people.stern.nyu.edu/churvich/Forecasting/Handouts/Scholes.pdf Brianegge,. VolatilitySmile. Retrieved November9th2016, from https://en.wikipedia.org/wiki/File:Volatility_smile.svg Citeas: Black-Scholes-Merton. Brilliant.org. 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