二次方风险最小化避险策略於附重设型选择权之权益指数年金之应用

風險管理學報第十一卷第一期2009年6月JournalofRiskManagementVol.11No.1June2009pp.95-117QuadraticRiskMinimizationHedgingStrategiesforRatchetOptionsinEquity-indexedAnnuities二次方風險最小化避險策略於附重設型選擇權之權益指數年金之應用鄭直夫（Chih-FuJheng）∗張智凱（Chih-KaiChang）∗∗AbstractThisstudyproposesaquadraticriskminimization(QRM)hedgingstrategyforthecompoundannualratchet(CAR)designinEquity-indexAnnuities.ComparedtohedgingstrategiesundertheBlackandScholesframework,theadvantageofQRMisthattheQRMsimultaneouslyconsidersthecapitalgain(orloss)andtheextrahedgingcost,fromthepracticalconstraintofdiscretehedgetrading,andminimizesthishedgingcostviatheLeastSquareMethod.ArecursivealgorithmisdevelopedfromtheCRRtreemodeltoovercomethepathdependencyoftheratchetoptionandtoimprovetheefficiencyoftheoriginalQRMhedgingstrategy.AMonteCarlosimulationmethodisusedtocomparetheeffectivenessoftheQRMhedgingstrategywiththedeltahedgingstrategy,basedonthreecriteria:initialhedgingcost,hedgingcost,andtransactioncost.ThenumericalresultshowsthatQRMoutperformsthedeltahedgingstrategyintermsofhedgingcostandtransitioncost,althoughQRMhasahigherinitialcostthanthedelta-hedgestrategyhas.Furthermore,empiricalinvestigationbasedontherecentbearishTaiwanstockmarketindicatesthattherecursiveQRMyieldslowercapitallossandhedgingcostthandelta-hedgedoes.Keywords:QuadraticRisk-Minimization,RatchetOption,Equity-IndexedAnnuities,andPathDependency.∗逢甲大學統計與精算研究所碩士生，Student,GraduateInstituteofStatisticsandActuarialScience∗∗逢甲大學統計與精算研究所助理教授，AssistantProfessor,GraduateInstituteofStatisticsandActuarialScienceQuadraticRiskMinimizationHedgingStrategiesforRatchetOptionsinEquity-indexedAnnuities96摘　要本文探討二次方風險最小化避險策略於複利重設型權益指數年金之應用。相較於Black-Scholes架構之下所使用的避險策略，二次方風險最小化避險策略的優點為：實務上避險的交易是無法連續進行的，並考慮在避險過程中，所衍生的資本利得（或利損），以及因無法完全避險所需的額外成本，透過最小平方法，將其避險成本最小化。結合二次方風險最小化避險策略與CRR二項樹模型，計算最佳化之避險策略。為解決複利重設型選擇權的報酬具有路徑相依的問題，本文推導出計算複利重設型選擇權避險策略之遞迴公式。此遞迴公式可以藉由納入未來金融市場之預期改善二次方風險最小化避險策略之效用。利用此遞迴公式，以蒙地卡羅模擬方法，以起始成本、避險成本與交易成本三種標準，比較二次方風險最小化避險策略與DeltaHedge兩種避險之優劣。模擬數值結果發現，雖然二次方風險最小化避險策略的起始成本較DeltaHedge略高，但二次方風險最小化避險策略的避險成本與交易成本顯著低於DeltaHedge。整體而言，二次方風險最小化避險策略優於DeltaHedge；近一步以近年來台灣股票市場之實證結果發現，本文所提供之二次方風險最小化避險策略與遞迴公式可以反映出較為長期之預測，較DeltaHedge能避免資本損失並減少避險成本與交易成本。ᙯᔣෟ：二次方風險最小化避險策略、複利重設型選擇權、權益指數年金、路徑相依。風險管理學報第十一卷第一期2009年ٛ6月971.IntroductionAnequity-indexedannuity(EIA)isafixedannuitythatearnsaminimumrateofinterestandoffersapotentialgainthatistiedtotheperformanceofanequityindex.EIAsenablepolicyholderstoreceiveanexcessreturninadditiontothepromisedpaymentsthataregovernedbytheguaranteedinterestrate.EIAsappealtoinvestorsbecausetheynotonlyoffersomeofthebenefitsunderlyingconventionalannuities;theysimultaneouslyofferparticipationintheequitymarketandlimitthedownsiderisk.IntheUnitedStates,accordingtotheAdvantageIndexProductSalesReportatAnnuitySpecs.com,EIAsalesvolumeexceededUS$6,000millionsincethethirdquarterof2004andreachedapeakofaboutUS$7,500millioninthesecondquarterof2005.Inthefirstquarterof2008,EIAsaleswereUS$5,776million,butEIAsaleswereUS$6,436millionduringthepreviousquarter.TherearemanyEIA-likefinancialinstrumentsinTaiwan,suchasStructuredNotes,CapitalGuaranteedNotes,Equity-linkedNotesandParticipationNotes.Inpractice,theprincipalofStructuredNotesisusedforbuyingfixedincomeproductslikezero-couponbondsforacertaindegreeofprincipalprotection,whiletheinterestisinvestedinderivativeslinkedtoequityindexes,foreignexchangerates,commodityfutures;thereturnoftheStructuredNotesdependsontheunderlyingperformanceoftheseinvestments.Duringthefirstmonthof2008,Taiwan’sStructuredNotessalesvolumereachedNT$50billion.Afterthe2008financialcrisis,bothTaiwan’sregulatoryauthoritiesandcapitalmarketinvestorsdirectedconsiderableattentiontotheinsolvencyandcreditissuesofStructuredNotes.AtypicalEIAguaranteesaminimumreturnonaportionoftheinitialamountinvested.Inadditiontothisguaranteedminimumpayment,theannuitantreceivessomebenefitfromtheappreciationofapredeterminedequityindex.ThereareseveralindexingmethodsforEIAs,suchas:point-to-point,compoundannualratchet,simpleannualratchetandhighwater.Thecompoundannualratchetisthemostpopulartype;itsannualyieldisresetannuallyaccordingtotheperformanceofthespecifiedindexanditsguaranteedminimuminterestrate.DuetotheexistenceofguaranteedEIAinterestrates,theparticipatingmechanismresemblesthatofratchetoptions.Therearemanysimilarinsurancecontractsthatincluderatchetoptions,suchasGuaranteedInvestmentContracts(GICs)andparticipatingpolicies.ThemostimportantdifferencebetweenthesefinancialinstrumentsandEIAsisthattheQuadraticRiskMinimizationHedgingStrategiesforRatchetOptionsinEquity-indexedAnnuities98excessreturnofEIAsisindexedaccordingtoaspecifiedequity,suchastheS&P500,whereastheexcessreturnsofparticipatingpoliciesorGICsdependontheissuers’equityperformancelevels.Therehasbeensomeresearchonratchetoptionvaluationinfinancialinstruments.LinandTan(2002)consideredthepricingofEIAsunderstochasticinterestrates.Tiong(2000)presentedpricingformulasforthreeofthemostcommonproductdesigns—thepoint-to-point,thecliquet,andthelookback—byemployinganEsschertransformmethod.GrosenandJørgensen(2000),Jensen,