MATLAB tutorial 2.6: Black Scholes model - Fluids at Brown

This model was later built out by Fischer Black and Myron Scholes to develop the Black--Scholes pricing model. Robert C. Merton is a famed American economist ... MATLABtutorial2.6:BlackScholesmodel BlackScholesmodel FischerBlack MyronScholes RobertC.Merton Anoptionscontractisafinancialinstrumentwhichgivestheownertheright, butnottherequirement,tobuyorsellstockonaparticulardate(the expirationdate)foraspecifiedprice(thestrikeprice).Acallcontractis thecontractthatprovidestheownertherighttopurchasestockatthestrike priceontheexpirationdate,andaputcontractallowstheownertosellstock atthestrikepriceontheexpirationdate.Thesecontractsarecommonlybought andsoldonpublicexchanges,andoptionsmarketsexistformostpublicly tradedstocks. Giventhatthecontracthasvaluetotheownerandplacesaliabilityonthe seller,itstandstoreasonthateachcontracthasafairvalueatanygiven pointintime.Inotherwords,ifweassumethatabuyerandasellerofan optionscontracthavethesameimperfectinformationaboutchangesinthestock price,thereshouldbesomepriceatwhichthebuyerdoesnotstandtohavea positiveexpectedgaingreaterthanthereturnprovidedbytheriskfree interestrate. TheBlackScholesmodel,alsoknownastheBlack--Scholes--Mertonmodel,isamodelofpricevariationovertimeoffinancialinstrumentssuchasstocksthatcan,amongotherthings,beusedtodeterminethepriceofaEuropeancalloption.ThemodelassumesthepriceofheavilytradedassetsfollowsageometricBrownianmotionwithconstantdriftandvolatility.Whenappliedtoastockoption,themodelincorporatestheconstantpricevariationofthestock,thetimevalueofmoney,theoption'sstrikeprice,andthetimetotheoption'sexpiry. TheBlackScholesmodelisoneofthemostimportantconceptsinmodernfinancialtheory.Itwasdevelopedin1973byFisherBlack,RobertMerton,andMyronScholesandisstillwidelyusednow.Itisregardedasoneofthebestwaysofdeterminingfairpricesofoptions.TheBlackScholesmodelrequiresfiveinputvariables:thestrikepriceofanoption,thecurrentstockprice,thetimetoexpiration,therisk-freerate,andthevolatility.Additionally,themodelassumesstockpricesfollowalognormaldistributionbecauseassetpricescannotbenegative.Moreover,themodelassumestherearenotransactioncostsortaxes;therisk-freeinterestrateisconstantforallmaturities;shortsellingofsecuritieswithuseofproceedsispermitted;andtherearenorisklessarbitrageopportunities. TheMertonmodelisananalysismodel–namedaftereconomistRobertC.Merton–usedtoassessthecreditriskofacompany'sdebt.AnalystsandinvestorsutilizetheMertonmodeltounderstandhowcapableacompanyisatmeetingfinancialobligations,servicingitsdebt,andweighingthegeneralpossibilitythatitwillgointocreditdefault.ThismodelwaslaterbuiltoutbyFischerBlackandMyronScholestodeveloptheBlack--Scholespricingmodel. RobertC.MertonisafamedAmericaneconomistandNobelMemorialPrizelaureate,whobefittinglypurchasedhisfirststockatage10.Later,heearnedaBachelorinScienceatColumbiaUniversity,aMastersofScienceatCaliforniaInstituteofTechnology(CalTech),andadoctorateineconomicsatMassachusettsInstituteofTechnology(MIT),wherehelaterbecomeaprofessoruntil1988.AtMIT,hedevelopedandpublishedgroundbreakingandprecedent-settingideastobeutilizedinthefinancialworld. PriortothedevelopmentoftheBlack-Scholesoptionspricingmodel,numericaltechniqueswere oftenusedtoestimatethefairpriceofanoptionscontract.Aswewillseelater,anoptionscontract isconsideredfairlypricedifthereisnowaytousesomecombinationofbuying/sellingstocksand optionstoearnariskfreeprofitgreaterthanthereturnonariskfreeasset. AcommontechniqueusedtomodelthestockmarketistoassumethatthepriceofastockfollowsGeometricBrownianMotionwithpositivedrift,meaningthatthepercent-wisemovementofthestockpriceisassumedtoberandomandnormallydistributedwithpositivemean.Itisalsoassumed thatthereexistsariskfreeinterestrate,meaningthatthereissomepercent-wisereturnthatcan beearnedoninvestedmoneywithnorisk.Generallythisisinterpretedtomeantheinterestrate onU.S.Treasurybonds,orsomeotherinstrumentwhichhasaverylowriskofdefault.Lastly,the stockisassumedtopaynodividend. TheBlack--Scholesmodelmakescertainassumptions: TheoptionisEuropeanandcanonlybeexercisedatexpiration. Nodividendsarepaidoutduringthelifeoftheoption. Marketsareefficient(i.e.,marketmovementscannotbepredicted). Therearenotransactioncostsinbuyingtheoption. Therisk-freerateandvolatilityoftheunderlyingareknownandconstant. Thereturnsontheunderlyingarenormallydistributed. Intherealworld,extremelylargemovementsinstockpricesaremuchmorec ommonthantheseassumptionswouldsuggest.Financialcrisesandlarge macroeconomicdisruptionscanoftencauseseveredownwardmovementsinstock marketsthatwouldbenearlyimpossibleifthemarketfollowedatrue GeometricBrownianMotion. Despitethisdifference,andafewothers,theaboveassumptionsarestill usedassimplifyingassumptionstodevelopthepricingmodelforoptions contracts. WedenotethestockpriceasafunctionoftimeS(t)andthe priceofthecallcontractasafunctionofboththestockpriceandtime V(S,t).Givenourassumptionthatthestockpricefollowsa GeometricBrownianMotion,wecanwritethat \[ {\textd}S=\muS\,{\textd}t+\sigmaS\,{\textd}X, \] whereXfollowsBrownianMotion.Intuitively,thisindicatesthat changesinthestockpricearerelatedtosomepositivedrifttermthatis proportionaltothestockprice,andtosomerandommovement,whoseaverage magnitudeisalsoproportionaltothestockprice. Supposewearetobuythecalloptionandsellthestockshort,meaningthat weborrowsharesofstockfromsomeoneelseandsellthematthecurrentmarket price,givingustheobligationtopurchasethesharesatalaterdateandr eturnthem.Ourpositionthushasanetvalueof \[ \Pi=V(S,t)-\DeltaS(t), \] wheredeltaistheamountofthestockthatweshort.(Intherealworld,it isnotpossibletoshortfractionsofastock,butthesevaluescanbescaled uptoverycloseintegerratios,soouranalysisisstillviable.) Fromthisequation,wecanget \[ {\textd}\Pi={\textd}V(S,t)-\Delta{\textd}S(t). \] ThenextsteprequiresustouseIto’sLemma,alemmathatisusedtocalculatethedifferentialof afunctionoftimeandastochasticfunction,whichisexactlywhathestatedtheoptionpriceto be.SeeEvans[5]formoreinformationonIto’sLemma.UsingIto’sLemma,wecanarriveat \[ {\textd}V=\frac{\partialV}{\partialS}\,{\textd}S+\frac{\partialV}{\partialt}\,{\textd}t+\frac{1}{2}\,\sigma^2S^2\frac{\partial^2V}{\partialS^2}\,{\textd}t. \] Pluggingintoourequationforthevalueofourportfolio,weget: \[ {\textd}\Pi=\frac{\partialV}{\partialS}\,{\textd}S+\frac{\partialV}{\partialt}\,{\textd}t+\frac{1}{2}\,\sigma^2S^2\frac{\partial^2V}{\partialS^2}\,{\textd}t-\Delta{\textd}S. \] Itwouldbetheoreticallypossibletoconstantlyadjustdeltasothat \[ \Delta=\frac{\partialV}{\partialS}. \] Wethereforecancelthefirstandlasttermtoarriveat \[ {\textd}\Pi=\frac{\partialV}{\partialS}\,{\textd}S+\frac{1}{2}\,\sigma^2S^2\frac{\partial^2V}{\partialS^2}\,{\textd}t. \] Becauseitshouldnotbepossibletoearnariskfreeprofitoffoftheoptionthatisgreaterthan theriskfreerateofreturn(whichwewilldenotehereasr),wealsohave \[ {\textd}\Pi=r\,\Pi\,{\textd}t=r\left(V-\DeltaS\right){\textd}t= r\left(V-\frac{\partialV}{\partialS}\,S\right){\textd}t \] becausetheriskfreeratewillearntheproductofthevalueofourportfolio, therateofreturn,andtimeelapsed.Therefore, \[ \frac{\partialV}{\partialt}\,{\textd}t+\frac{1}{2}\,\sigma^2S^2 \frac{\partial^2V}{\partialS^2}\,{\textd}t=r\left(V- \frac{\partialV}{\partialS}\,S\right){\textd}t. \] Upondroppingdt,itfollows \[ \frac{\partialV}{\partialt}+\frac{1}{2}\,\sigma^2S^2\frac{\partial^2V}{\partialS^2}=r\left(V-\frac{\partialV}{\partialS}\,S\right). \] ThisgivestheBlack--Scholesequation: \[ \frac{\partialV}{\partialt}+\frac{1}{2}\,\sigma^2S^2\frac{\partial^2V}{\partialS^2}+rS\,\frac{\partialV}{\partialS}-r\,V=0. \] ThepriceofanoptionV(S,t)isdefinedfor0K, thenthecontractisworthS-K.However,ifS< K,thenthecontractisworthlessbecausethereisnopointinexercising yourrighttobuyatKwhenyoucanjustbuyatthemarketprice S,whichisless.Thesecondconditionstatesthatthecallisworthless ifthestockpriceis0.Becauseweassumedthatthestockwouldfollow GeometricBrownianMotion,apriceof0wouldimplythatthepricewouldstay at0forever,sincemovementsareproportionalinmagnitudetothestockprice. Thereforethereisnovaluetoacallcontractifthepriceofthestockis0 (assumingthestrikepriceispositive). Weapplythetransformation\(u=V\,e^{−rt}\) (or\(V=u\,e^{rt}\))andaccordingly \(e^{rt}\frac{\partialu}{\partialS}= \frac{\partialV}{\partialS}\)and \(e^{rt}\frac{\partial^2u}{\partialS^2}= \frac{\partial^2V}{\partialS^2}\) totheBlack--Scholesequation \[ \frac{\partialV}{\partialt}+\frac{1}{2}\,\sigma^2S^2\frac{\partial^2V}{\partialS^2}+rS\,\frac{\partialV}{\partialS}-r\,V=0. \] toobtain \[ r\,e^{rt}u+e^{rt}\frac{\partialu}{\partialt}+\frac{1}{2}\,\sigma^2S^2 e^{rt}\frac{\partial^2V}{\partialS^2}+rS\,e^{rt}\frac{\partialu}{\partialS}-r\,e^{rt}u=0. \] Thisissimplifiedto \[ e^{rt}\frac{\partialu}{\partialt}+\frac{1}{2}\,\sigma^2S^2 e^{rt}\frac{\partial^2V}{\partialS^2}+rS\,e^{rt}\frac{\partialu}{\partialS}=0, \] whichuponeliminatingthecommonexponentialmultipleyields \[ \frac{\partialu}{\partialt}+\frac{1}{2}\,\sigma^2S^2 \frac{\partial^2V}{\partialS^2}+rS\,\frac{\partialu}{\partialS}=0, \] Fromhere,wecanattempttomakethesubstitutions \(S=e^y\)and \(t=T-\tau\)totransform u(S,t)tou(y,τ). Thisshouldlikewisetransform \(\frac{\partial}{\partialS}\)to \(\frac{1}{S}\,\frac{\partial}{\partialy}\)and \(\frac{\partial}{\partialt}\)to \(-\frac{\partial}{\partial\tau}.\) Wecanalsoderive \[ \frac{\partial}{\partialS^2}=-\frac{1}{S^2}\,\frac{\partial}{\partialy} +\frac{1}{S}\,\frac{\partial}{\partialS}\,\frac{\partial}{\partialy}= -\frac{1}{S^2}\,\frac{\partial}{\partialy} +\frac{1}{S^2}\,\frac{\partial}{\partialy^2}. \] Makingthesesubstitutionsandgroupingthe \(\frac{\partialu}{\partialy}\)terms,we arriveat \[ -\frac{\partialu}{\partial\tau}+\frac{1}{2}\,\sigma^2 \frac{\partial^2u}{\partialy^2}+\left(r-\frac{1}{2}\,\sigma^2\right) \frac{\partialu}{\partialy}=0. \] Weuseonelasttransformation, \[ z=y+\left(r-\frac{1}{2}\,\sigma^2\right)\overline{\tau}\qquad \mbox{and}\qquad\overline{\tau}=\tau, \] andthecorrespondingrelationships \(\frac{\partial}{\partialy}=\frac{\partial}{\partialz} \)and \[ \frac{\partial}{\partial\overline{\tau}}=\frac{\partial}{\partial\tau}\, \frac{\partial\tau}{\partial\overline{\tau}}+\frac{\partial}{\partialy}\, \frac{\partialy}{\partial\overline{\tau}}= \frac{\partial}{\partial\tau}-\left(r-\frac{1}{2}\,\sigma^2\right) \frac{\partial}{\partialy}. \] WecansubstitutealltheseequationsintoBlack--Scholesequationtoobtain \[ -\frac{\partialu}{\partial\overline{\tau}}-\left(r-\frac{1}{2}\, \sigma^2\right)\frac{\partialu}{\partialz}+ \frac{1}{2}\,\sigma^2\frac{\partial^2u}{\partialz^2}+ \left(r-\frac{1}{2}\, \sigma^2\right)\frac{\partialu}{\partialz}=0, \] whichcanbesimplifiedfurthertotheheatequation: \[ \frac{\partialu}{\partial\overline{\tau}}=\frac{1}{2}\,\sigma^2 \frac{\partial^2u}{\partialz^2} \] Theinitialconditionistransformedto \[ u(z,\tau)=e^{-r(T-\tau)}\,V\left(e^{z-(r-\sigma^2/2)},T-\tau \right)\qquad\Longrightarrow\qquadu(z,0)=e^{-rT}\,V\left(e^z,T \right)=e^{-rT}\,\max\left(0,e^z-K\right). \] Ourfinalresultissimplytheheatdiffusionequation \(\frac{\partialu}{\partialt}=k\, \frac{\partial^2u}{\partialx^2}\)subjecttotheinitialcondition u(x,0)=φ(x).Itssolutionisknowntobe \[ u(x,t)=\frac{1}{\sqrt{4\pikt}}\,\int_{-\infty}^{\infty} e^{-(x-y)^2/(4kt)}\,\varphi(y)\,{\textd}y. \] Ifweplugourvaluesintothissolution,anddosomealgebratoreplacesome ofthetransformedvariableswiththeoriginals(usingdifferentvariable namestoavoidconfusionwiththeonesusedbefore),wehave \[ V(S,t)=\frac{e^{-r\tau}}{\sqrt{2\pi\sigma^2\tau}}\,\int_{-\infty}^{\infty} \exp\left\{-\frac{\left(r-\sigma^2/2\right)\tau+\ln(S)-\eta}{2\sigma^2\tau}\right\}\max\left(0,e^{\eta-K}\right){\textd}\eta, \] whichevaluatestotheBlack--ScholesFormula \[ V(S,\tau)=S\,\Phi\left(d_1(S,\tau)\right)-K\,e^{-r\tau}\Phi \left(d_2(S,\tau)\right), \] where \[ \begin{split} d_1(S,\tau)&=\frac{1}{\sigma\sqrt{\tau}}\left[\ln\frac{S}{K}+ \left(r+\frac{1}{2}\,\sigma^2\right)\tau\right],\\ d_2(S,\tau)&=\frac{1}{\sigma\sqrt{\tau}}\left[\ln\frac{S}{K}+ \left(r-\frac{1}{2}\,\sigma^2\right)\tau\right], \end{split} \] whereΦisthenormalcumulativedistributionfunctionandτ=T-t. UsingthesolutiontotheBlack-Scholesequation,wecansimulatethepriceof acallorputcontractexpiringonSept1,2018withvariousstrikeprices, startingayearbeforethatonSept1,2017.matlabcodeusedforthe approximationisshownbelow: functionbs K=2600:100:3100;%strikeprices n=numel(K); sigma=xlsread('VIX','sheet');%volatility S=xlsread('SP500','sheet'); %Callprices figure holdon fori=1:n cblackscholes(K(i),false,sigma,S) end labels('Call'); legend('location','NorthWest') %Putprices figure holdon fori=1:n cblackscholes(K(i),true,sigma,S) end labels('Put'); end functioncblackscholes(K,isput,sigma,S) %constants r=0.02;%riskfreeinterestrate T=1;%day n=length(S); t=linspace(0,1,n); call(n)=0;%memoryallocation fori=1:n tau=T-t(i); d1=1/(sigma(i)*sqrt(tau))*(log(S(i)/K)+(r+sigma(i)^2/2)*tau); d2=d1-sigma(i)*sqrt(tau); call(i)=N(d1)*S(i)-N(d2)*K*exp(-r*tau);%Black-Scholesforumla ifisput call(i)=call(i)-S(i)+K*exp(-r*tau); end end plot(t,call) end functionwt=N(d)%CDFofnormaldistribution pd=makedist('Normal');%Makeprobabilitydistribution wt=cdf(pd,d); end functionlabels(cp) legend('K=2600','K=2700','K=2800','K=2900','K=3000','K=3100') xlabel('TimeElapsed,years') ylabel([cp,'Price,$']) title(['Black-ScholesSimulated',cp,'PricesforS&P500,8/17-8/18'],'fontsize',12) holdoff end UsingthiscodeanddataVIXandSP500,wegeneratethefollowinggraphs: BlackScholessimulatedPutPricesS&P5008/17--8/18 BlackScholessimulatedCallPricesS&P5008/17--8/18 References Black,FischerandScholes,Myron(1973)."ThePricingofOptionsandCorporateLiabilities".JournalofPoliticalEconomy.81(3):637--654.doi:10.1086/260062 Bodie,Zvi,Kane,Alex,andMarcus,Alan,Investments,11thEdition,McGraw-HillEducation,2018. 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Merton,RobertC.(1973)."TheoryofRationalOptionPricing".BellJournalofEconomicsandManagementScience.TheRANDCorporation.4(1):141–183.doi:10.2307/3003143. Stecher,Michael,ConvertingtheBlack-ScholesPDEtotheheatequation Stehlíková,Beáta,Black-Scholesmodel:Derivationandsolution Yalincak,OrhunHakan.CriticismoftheBlack-ScholesModel:ButWhyisItStillUsed?:(The AnswerisSimplerthantheFormula).NewYorkUniversity.NewYork,NY,2005.