A Path Integral Approach to Option Pricing with St

arXiv:cond-mat/9708178v122Aug1997APathIntegralApproachtoOptionPricingwithStochasticVolatility:SomeExactResultsBelalE.Baaquie1DepartmentOfPhysics,NationalUniversityofSingapore,KentRidge,Singapore119260AbstractTheBlack-Scholesformulaforpricingoptionsonstocksandothersecuritieshasbeengener-alizedbyMertonandGarmantothecasewhenstockvolatilityisstochastic.Thederivationofthepriceofasecurityderivativewithstochasticvolatilityisreviewedstartingfromthefirstprinciplesoffinance.TheequationofMertonandGarmanisthenrecastusingthepathintegrationtechniqueoftheoreticalphysics.ThepriceofthestockoptionisshowntobetheanalogueoftheSchrodingerwavefuctionofquantummechacnicsandtheexactHamiltonianandLagrangianofthesystemisobtained.TheresultsofHullandWhitearegeneralizedtothecasewhenstockpriceandvolatilityhavenon-zerocorrelation.Someexactresultsforpricingstockoptionsforthegeneralcorrelatedcasearederived.1Email:[e-mail]mofpricingtheEuropeancalloptionhasbeenwell-studiedstartingfromthepioneeringworkofBlackandScholes[1],[2].TheresultsofBlackandScholesweregeneral-izedbyMerton,Garman[3]andothersforthecaseofstochasticvolatilityandtheyderivedapartialdifferentialequationthattheoptionpricemustsatisfy.ThemethodsoftheoreticalphysicshavebeenappliedwithsomesuccesstotheproblemofoptionpricingbyBouchaudetal[4].Analyzingtheproblemofoptionpricingfromthepointofviewofphysicsbringsawholecollectionofnewconceptstothefieldofmathematicalfinanceaswellasaddstoitasetofpowerfulcomputationaltechniques.Thispaperisacontinuationofapplyingthemethodologyofphysics,inparticular,thatofpathintegralquantummechanics,tothestudyofderivatives.Itshouldbenotedthat,unlikethepaperbyBouchaudandSornette[4]inwhichthepricingofderivativesisobtainedbytechniqueswhichgobeyondtheconventionalapproachinfinance,thepresentpaperisbasedontheusualprinciplesoffinance.Inparticular,acontinuoustimerandomwalkisassumedforthesecurityandarisk-freeportfolioisusedtoderivethederivativepricingequation;thesearereviewedinSecII.Themainfocusofthispaperistoapplythecomputationaltoolsofphysicstothefieldoffinance;themorechallengingtaskofradicallychangingtheconceptualframeworkoffinanceusingconceptsfromphysicsisnotattempted.AnexplicitanalyticalsolutionforthepricingoftheEuropeancall(orput)optionforthecaseofstochasticvolatilityhassofarremainedanunsolvedproblem.HullandWhite[5]studiedthisproblemandobtainedaseriessolutionforthecasewhenthecorrelationofstockpriceandvolatility,namelyr,isequaltozero,i.e.ρ=0(uncorrelated).HullandWhite[5]alsoobtainedanalgorithmtonumericallyevaluatetheoptionpriceusingMonteCarlomethodsforthecaseofρ6=0.Extensivenumericalstudiesofoptionpricingforstochasticvolatilitybasedonthealgo-rithmofFinucane[6]havebeencarriedoutbyMillsetal[7].ThemainfocusandcontentofthispaperistostudytheproblemofstochasticvolatilityfromthepointofviewoftheFeynmanpathintegral[9].TheFeynman-KacformulafortheEuropeancalloptioniswellknown[2,8].Therehavealsobeensomeapplicationsofpathintegralsinthestudyofoptionpricing[10,11].Inthispaperthepathintegralapproachtooptionpricingisfirstdiscussedinitsgeneralityandthenappliedtotheproblemofstochastic2volatility.TheadvantageofrecastingtheoptionpricingproblemasaFeynmanpathintegralisthatthisallowsforanewpointofviewandwhichleadstonewwaysofobtainingsolutionswhichareexact,approximateaswellasnumerical,tothethepricingofoptions.InSectionIIthedifferentialequationofMertonandGarman[3]isderivedfromtheprinciplesoffinance.InSectionIIIthisequationisrecastintheformalismofquantummechanics.InSectionIVadiscretetimepathintegralexpressionisderivedfortheoptionpricewithstochasticvolatilitywhichgeneralizestheFeynman-Kacformula.InSectionVthepathintegrationoverthestochasticstockpriceisperformedexplicitlyandacontinuoustimepathintegralisthenobtained.AndlastlyinSectionVIsomeconclusionsaredrawn.IISecurityDerivativewithStochasticVolatilityWereviewtheprinciplesoffinancewhichunderpinthetheoryofsecurityderivatives,andinparticularthatofthepricingofoptions.WewillderivetheresultsnotusingtheusualmethodusedintheoreticalfinancebasedonIto-calculus(whichwewillreviewforcompleteness),butinsteadfromtheLangevinstochasticdifferentialequation.Hencewestartfromfirstprinciples.Asecurityisanyfinancialinstrumentwhichistradedinthecapitalsmarket;thiscouldbethestockofacompany,theindexofastockmarket,governmentbondsetc.Asecurityderivativeisafinancialinstrumentwhichisderivedfromanunderlyingsecurityandwhichisalsotradedinthecapitalsmarket.Thethreemostwidelyusedderivativesareoptions,futuresandforwards;morecomplicatedderivativescanbeconstructedoutofthesemorebasicderivartives.Inthispaper,weanalyzetheoptionofanunderlyingsecurityS.AEuropeancalloptiononSisafinancialinstrumentwhichgivestheowneroftheoptiontobuyornottobuythesecurityatsomefuturetimeT¿tforthestrikepriceofK.Attimet=T,whentheoptionmaturesthevalueofthecalloptionf(T)isclearlygivenbyf(T,S(T))=S(T)−K,S(T)K0,S(T)K(1a)=g(S)(1b)3Theproblemofoptionpricingisthefollowing:giventhepriceofthesecurityS(t)attimet,whatshouldbethepriceoftheoptionfattimetT?Clearlyf=f(t,S(t),K,T).Thisisafinalvalueproblemsinceth