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1、ANumericalSchemefortheImpulseControlFormulationforPricingVariableAnnuitieswithaGuaranteedMinimumWithdrawalBenefit(GMWB)ZhuliangChen∗PeterA.Forsyth†July9,2007AbstractInthispaper,weoutlineanimpulsestochasticcontrolformulationforpricingvariablean-nuitieswit
2、haGuaranteedMinimumWithdrawalBenefit(GMWB)assumingthepolicyholderisallowedtowithdrawfundscontinuously.WedevelopasinglenumericalschemeforsolvingtheHamilton-Jacobi-Bellman(HJB)variationalinequalitycorrespondingtotheimpulsecontrolproblem,andforpricingrealis
3、ticdiscretewithdrawalcontracts.Weprovetheconvergenceofourschemetotheviscositysolutionofthecontinuouswithdrawalproblem,providedastrongcomparisonresultholds.Theconvergencetotheviscositysolutionisalsoprovedforthediscretewithdrawalcase.Numericalexperimentsa
4、reconducted,whichshowaregionwheretheoptimalcontrolappearstobenon-unique.Keywords:Impulsecontrol,GMWB,finitedifference,viscositysolution.AMSClassification:65N06,93C20Acknowledgment:ThisworkwassupportedbytheNaturalSciencesandEngineeringRe-searchCouncilofCana
5、da,andbyaMorganStanleyEquityMarketMicrostructureResearchGrant.Theviewsexpressedhereinaresolelythoseoftheauthors,andnotthoseofanyotherpersonorentity,includingMorganStanley.1IntroductionVariableannuitieswithaGuaranteedMinimumWithdrawalBenefit(GMWB)areextre
6、melypop-ularsincethesecontractsprovideinvestorswiththetax-deferredfeatureofvariableannuitiesaswellastheadditionalbenefitoftheguaranteedminimumpayment.In2004,sixty-ninepercentofallvariableannuitycontractssoldintheUSincludedaGMWBoption[4].AGMWBcontractinvo
7、lvespaymentofalumpsumtoaninsurancecompany.Thislumpsumistheninvestedinriskyassets.Theholderofthiscontractmaywithdrawuptoaspecifiedamountineachyearforthelifeofthecontract,regardlessoftheperformanceoftheriskyasset.Theholdermayalsowithdrawmorethanthespecified
8、amount,subjecttocertainpenaltiesandconditionson∗DavidR.CheritonSchoolofComputerScience,UniversityofWaterloo,WaterlooON,CanadaN2L3G1e-mail:z4chen@uwaterloo.ca†DavidR.CheritonSchoolofComputerScience,UniversityofWaterloo,WaterlooON,
