Analysis of the stability of the linear boundary c

ANALYSISOFTHESTABILITYOFTHELINEARBOUNDARYCONDITIONFORTHEBLACK-SCHOLESEQUATION.H.WINDCLIFFy,P.A.FORSYTHz,ANDK.R.VETZALxAbstract.Thelinearasymptoticboundarycondition,i.e.assumingthatthesecondderivativeofthevalueofthederivativesecurityvanishesastheassetpricebecomeslarge,iscommonlyusedinpractice.Toourknowledge,therehavebeennorigorousstudiesofthestabilityofthesemethods,despitethefactthatthediscretematrixequationsobtainedusingthisboundaryconditionlosesdiagonaldominanceforlargetimesteps.Inthispaper,wedemonstratethatthediscreteequationsobtainedusingthisboundaryconditionsatisfynecessaryconditionsforstabilityforanitedierencediscretization.Computationalexperimentsalsoshowthatthisboundaryconditionsatisessucientconditionsforstabilityaswell.Keywords:Asymptoticboundarycondition,stability,nitedierence,Black-ScholesequationJanuary14,20031.Motivation.WhensolvingoptionpricingPDEssuchastheBlack-Scholesequationnumerically,manyauthors[18,17,10]haverecommendedalinearasymptoticboundarycondition(thatthesecondderivativeoftheoptionvaluewithrespecttotheunderlyingassetvaluebezero)astheassetpricebecomeslarge.Althoughthisboundaryspecicationisoftenapplied,toourknowledgetherehavebeennorigorousstudiesofthestabilityofthistechnique.Lookingattheformofthediscretematrixequationsobtainedusingthisboundarycondition,theresultingdiscreteequationslosediagonaldominanceforlargetimestepsandtheusualargumentscannotbeappliedtoguaranteeunconditionalstability.Inordertodeterminetherangesofparameters(risk-freerate,volatility,etc.)forwhichthisasymptoticboundaryconditioncouldcauseinstability,wederivenecessaryconditionsforthestabilityofthediscreteequationsbasedonthespectrumofthematrixrepresentingthespatialdiscretization.Somewhatsurprisingly,wendthatanitedierence(FD)discretizationalwayssatisesthesenecessaryconditionsforstability,despitethefactthatthematrixequationsarenotunconditionallydiagonallydominant.Theeigenvaluescanbeusedtodeterminenecessaryconditionsforstabilitybutareknowntobeunreliablefordeterminingsucientconditionsforstability.FornitedimensionalproblemsanalysisofthespectrumcanleadtosucientconditionsbutinthePDEcontext,thesizeofthematrixbecomesunboundedasthegridisrened.Inourcasethematrixisnon-symmetricandnon-normalwhichfurthercomplicatesmatters.Fornon-normalmatrices,counterexamplescanbegivenwhere,eveniftheeigenvaluesarelessthanoneinmagnitude,instabilityresultsasthedimensionofthematrixbecomeslarge(see[11,7,9,16]).Forsomevaluesofthemarketparametersweareabletoshowthatsucientconditionsforstabilityaresatisedusingnumericalrangearguments[7,8,15,3].Inothercases,theseargumentscannotbeappliedandwefollow[2]anddemonstratethatthediscretetimesteppingoperatorispower-bounded.ThisworkwassupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanada,theSocialSciencesandHumanitiesResearchCouncilofCanadaandRBCFinancialGroup.yDepartmentofComputerScience,UniversityofWaterloo,WaterlooON,CanadaN2L3G1,[e-mail]cience,UniversityofWaterloo,WaterlooON,CanadaN2L3G1,[e-mail],xCentreforAdvancedStudiesinFinance,UniversityofWaterloo,WaterlooON,CanadaN2L3G1,[e-mail],P.A.FORSYTHandK.R.VETZAL2.ProblemFormulation.WewillconsiderthestandardBlack-Scholesequa-tion,whichcanbewrittenas:Vt+(rq)SVS+2S22VSSrV=0;(2.1)whereV(S;t)representsthevalueofthederivativesecurity,Sisthevalueoftheunderlyingsecurity,ristherisk-freeinterestrate,qisthecontinuousdividendyieldand,whichmaybeafunctionofSandt,isthevolatilityoftheunderlyingasset.Oncethecontractuallydenedpayoofthederivativesecurity,V(S;T)=g(S),isspecied,equation(2.1)canbesolvedbackwardsintimefromthematuritydateofthecontract,t=T,tothepresenttime,t=0,inordertoobtainthecurrentvalueofthecontract.Theoriginalproblem(2.1)isposedonthedomain[0;1).TheboundaryconditionatS=0isobtainedsimplybysettingS=0inequation(3.1).Thisresultsinthespecication:Vt(0;t)=rV(2.2)atthelowerboundary.Ofcourse,whenusinganimplicittypeofnumericalscheme,wemusttruncatethisdomainto[0;Smax].Consequently,itisalsonecessarytoimposeaboundaryconditionatS=Smax.Iftheerrormadeintheapproximationofthisboundaryconditionisboundedthen,byextendingthecomputationaldomain,itispossibletomakethenear-elderrorarbitrarilysmall.In[6]theauthorsshowthattherequiredsizeofthecomputationaldomainisproportionaltoepT,whereisthefar-elderror.Inordertobeabletoutilizesmallcomputationaldomains,itisimportantthattheerrorintheapproximationoftheboundaryconditionbeassmallaspossible.Forpath-dependentoptionstheusualruleofthumbforvanillaoptionsoftrun-catingthedomainat\threeorfourtimestheexercisepricequotedin[6]isnotalwayssucienttoensureaccurateresults.Thisisbecausethenear-eldsolutionmaydependonfar-elddatathroughthecontractuallydenedjumpconditions.Anexampleofsuchasituationisinthefullthree-dimensionalnumericalvaluationofmultipleshoutoptionsdescribedin[20]wheretherequiredsizeofthecomputationaldomaingrowsexponentiallywiththenumberofshoutopportunities.Somepopu-larinsuranceproductsoeredinCanada[19]canoerthe