hedging costs for variable annuities

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1、HedgingCostsforVariableAnnuitiesAPDERegime-SwitchingApproachbyParsiadAzimzadehAthesispresentedtotheUniversityofWaterlooinfulllmentofthethesisrequirementforthedegreeofMasterofMathematicsinComputerScienceWaterloo,Ontario,Canada,2013cParsiadAzimzadeh2013Iherebydec

2、larethatIamthesoleauthorofthisthesis.Thisisatruecopyofthethesis,includinganyrequirednalrevisions,asacceptedbymyexaminers.Iunderstandthatmythesismaybemadeelectronicallyavailabletothepublic.iiAbstractAgeneralmethodologyisdescribedinwhichpolicyholderbehaviourisdec

3、oupledfromthepricingofavariableannuitybasedonthecostofhedgingit,yieldingtwosequencesofweaklycoupledsystemsofpartialdierentialequations(PDEs):thepricingandutilitysystems.Theutilitysystemsareusedtogeneratepolicyholderwithdrawalbehaviour,whichisinturnfedintothepri

4、cingsystemsasameanstodeterminethecostofhedgingthecontract.Thisapproachallowsustoincorporatetheeectsofutility-basedpricingandfactorssuchastaxation.Asacasestudy,weconsidertheGuaranteedLifelongWithdrawalandDeathBenets(GLWDB)contract.Thepricingandutilitysystemsfor

5、theGLWDBarederivedundertheassumptionthattheunderlyingassetfollowsaMarkovregime-switchingprocess.AnimplicitPDEmethodisusedtosolvebothsystemsintandem.Weshowthatforalargeclassofutilityfunctions,thetwosystemspreservehomogeneity,allowingustodecreasethedimensionalityo

6、fsolutions.WealsoshowthattheassociatedcontrolfortheGLWDBisbang-bang,underwhichtheworkrequiredtocomputetheoptimalstrategyissignicantlyreduced.Weextendthisresulttoprovidethereaderwithsucientconditionsforabang-bangcontrolforageneralvariablean-nuitywithacountablen

7、umberofevents(e.g.discontinuouswithdrawals).Homo-geneityandbang-bangnessyieldsignicantreductionsincomplexityandallowustorapidlygeneratenumericalsolutions.Resultsarepresentedwhichdemonstratethesensitivityofthehedgingexpensetovariousparameters.Thecostlynatureofth

8、edeathbenetisdocumented.Itisalsoshownthatforatypicalcontract,thefeerequiredtofundthecostofhedgingcalculatedundertheassumptionthatthepolicyholderwithdrawsatthecontrac