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ZeroLowerBoundTermStructureModeling AppliedQuantitativeFinanceseries Applied Quantitative Finance is a new series developed to bring readers the very latest market tested tools, techniques and developments in quantitative finance. Writtenforpractitionerswhoneedtounderstandhowthingswork“onthefloor,” theserieswilldeliverthemostcutting-edgeapplicationsinareassuchasassetpric- ing,riskmanagementandfinancialderivatives.Althoughwrittenwithpractitioners inmind,thisserieswillalsoappealtoresearchersandstudentswhowanttoseehow quantitativefinanceisappliedinpractice. Alsoavailable ChrisKenyon,RolandStamm DISCOUNTING,LIBOR,CVAANDFUNDING InterestRateandCreditPricing MarcHenrard INTERESTRATEMODELLINGINTHEMULTI-CURVEFRAMEWORK Foundations,EvolutionandImplementation AdilReghai QUANTITATIVEFINANCE BacktoBasicPrinciples IgnacioRuiz XVADESKS:ANEWERAFORRISWKMANAGEMENT Understanding,BuildingandManagingCounterpartyandFundingRisk LeoKrippner ZEROLOWERBOUNDTERMSTRUCTUREMODELING APractitioner’sGuide Zero Lower Bound Term Structure Modeling A Practitioner’s Guide Leo Krippner ZEROLOWERBOUNDTERMSTRUCTUREMODELING Copyright©LeoKrippner,2015. Softcover reprint of the hardcover 1st edition 2015 978-1-137-40832-7 Allrightreserved. Firstpublishedin2015by PALGRAVEMACMILLAN® intheUnitedStates—adivisionofSt.MartinsPressLLC, 175FifthAvenue,NewYork,NY10010. WherethisbookisdistributedintheUK,Europeandtherestoftheworld, thisisbyPalgraveMacmillan,adivisionofMacmillanPublishersLimited, registeredinEngland,companynumber785998,ofHoundmills, Basingstoke,HampshireRG216XS. PalgraveMacmillanistheglobalacademicimprintoftheabovecompanies andhascompaniesandrepresentativesthroughouttheworld. Palgrave®andMacmillan®areregisteredtrademarksintheUnitedStates, theUnitedKingdom,Europeandothercountries. ISBN 978-1-349-68123-5 ISBN 978-1-137-40182-3 (eBook) DOI 10.1057/9781137401823 LibraryofCongressCataloging-in-PublicationData Krippner,Leo. Zerolowerboundtermstructuremodeling:apractitioner’s guide/LeoKrippner. pagescm.—(Appliedquantitativefinance) Includesbibliographicalreferencesandindex. ISBN1–137–40832–4 1.Finance–Mathematicalmodels.2.Structuralequationmodeling. 3.Economics,Mathematical.I.Title. HG176.5.K752015 339.5’30151953–dc23 2014027511 AcataloguerecordofthebookisavailablefromtheBritishLibrary. DesignbyNewgenKnowledgeWorks(P)Ltd.,Chennai,India. Firstedition:January2015 10 9 8 7 6 5 4 3 2 1 Tomyparents,JimandTeresaKrippner,andmypartner,IrisClaus. Contents ListofFigures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi SelectedListofNotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxiii ClassificationandAbbreviationsforTermStructureModels . . . . . . . . . . . .xxvii 1 Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Chapteroverview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Suggestedreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Group1:Generalmonetarypolicyreaders . . . . . . . . . . . . . 4 1.2.2 Group2:Generalfinancialmarketreaders . . . . . . . . . . . . . 4 1.2.3 Group3:Termstructuremodelers . . . . . . . . . . . . . . . . . . 4 1.3 Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Availabilityofresultsandcode. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Referencestotheliterature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Otherpreliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 ANewFrameworkforaNewEnvironment . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Monetarypolicy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Pre-GFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Post-GFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Termstructuremodeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Pre-GFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Post-GFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Shadow/ZLBtermstructuremodels . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 ZLBmechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Optionstoholdphysicalcurrency. . . . . . . . . . . . . . . . . . . 36 2.4 Monetarypolicyrevisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 AlternativeZLBmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.1 Square-roottermstructuremodels . . . . . . . . . . . . . . . . . . 42 2.5.2 Log-normaltermstructuremodels . . . . . . . . . . . . . . . . . . 43 2.5.3 QuadraticGaussiantermstructuremodels. . . . . . . . . . . . . 43 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 viii Contents 3 GaussianAffineTermStructureModels . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 GATSMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.1 GATSMspecification. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.2 GATSMdynamicsandrelatedcalculations . . . . . . . . . . . . . 48 3.1.3 GATSMtermstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 GATSMestimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Kalmanfilterequationsandrelatedparameters. . . . . . . . . . 53 3.2.2 Partialestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.3 Fullestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Workedexample:ANSM(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1 ANSM(2)Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.2 ANSM(2)Termstructure. . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.3 PartialANSM(2)estimationwiththeKalmanfilter . . . . . . . 66 3.3.4 FullANSM(2)estimationwiththeKalmanfilter . . . . . . . . . 70 3.4 OtherGATSMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.4.1 ANSM(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4.2 Higher-orderANSMs . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.4.3 Non-arbitrage-freeANSMs . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.4 StationaryGATSMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4.5 StationaryGATSMswithrepeatedeigenvalues . . . . . . . . . . 90 3.5 Empiricalapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.5.1 Yieldcurvedataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.5.2 ANSM(2)results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.5.3 ANSM(3)results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.6 Alternativeestimationmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.6.1 Estimationusingforwardrates . . . . . . . . . . . . . . . . . . . . . 100 3.6.2 Estimationusingbondprices . . . . . . . . . . . . . . . . . . . . . . 103 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 KrippnerFrameworkforZLBTermStructureModeling. . . . . . . . . . . . . 105 4.1 K-AGMexposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1.1 K-AGMIntuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.2 K-AGMoptioneffect . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.1.3 K-AGMforwardrates . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1.4 ObservationsontheK-AGMframework . . . . . . . . . . . . . . 112 4.1.5 Comparisonwithrelatedresultsintheliterature . . . . . . . . . 114 4.1.6 K-AGMinterestratesandbondprices . . . . . . . . . . . . . . . . 115 4.2 K-AGMestimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.2.1 K-AGMsandnonlinearKalmanfilters. . . . . . . . . . . . . . . . 117 4.2.2 Partialestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.2.3 Fullestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.3 Workedexample:K-ANSM(2). . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.3.1 K-ANSM(2)specification . . . . . . . . . . . . . . . . . . . . . . . . 126 Contents ix 4.3.2 K-ANSM(2)shadowtermstructure . . . . . . . . . . . . . . . . . 128 4.3.3 K-ANSM(2)ZLBtermstructure . . . . . . . . . . . . . . . . . . . . 129 4.3.4 K-ANSM(2)Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.4 OtherK-AGMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.4.1 K-ANSM(3).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.4.2 Higher-orderK-ANSMs . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.4.3 Non-arbitrage-freeK-ANSMs. . . . . . . . . . . . . . . . . . . . . . 136 4.4.4 StationaryK-AGMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.5 Empiricalapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.5.1 K-ANSM(2)results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.5.2 K-ANSM(3)results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.6 Alternativeestimationmethods . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.6.1 Estimationusingforwardrates . . . . . . . . . . . . . . . . . . . . . 147 4.6.2 Estimationusingbondprices . . . . . . . . . . . . . . . . . . . . . . 152 4.6.3 IterativeestimationusingGATSMs . . . . . . . . . . . . . . . . . . 155 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5 BlackFrameworkforZLBTermStructureModeling . . . . . . . . . . . . . . . 159 5.1 TheB-AGMframeworkprinciples . . . . . . . . . . . . . . . . . . . . . . . . 160 5.1.1 InitialcomparisonoftheB-AGMandK-AGM frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.2 B-AGMimplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.2.1 Customizedcalculationmethods . . . . . . . . . . . . . . . . . . . 168 5.2.2 Finitedifferencemethods. . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2.3 Latticemethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.2.4 MonteCarlomethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.3 B-AGMMonteCarloimplementations . . . . . . . . . . . . . . . . . . . . . 172 5.3.1 B-AGMMonteCarlospecification . . . . . . . . . . . . . . . . . . 173 5.3.2 ComputingtimeforB-AGMMonteCarlosimulations . . . . 178 5.3.3 Antitheticsampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.3.4 Otherpotentiallyusefulstandardspeed-upmethods . . . . . . 181 5.3.5 Workedexample:B-ANSM(2) . . . . . . . . . . . . . . . . . . . . . 181 5.3.6 OtherB-AGMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.4 K-AGMasacontrolvariateforB-AGMMonteCarlosimulations. . . 185 5.4.1 Definingthecontrolvariate . . . . . . . . . . . . . . . . . . . . . . . 186 5.4.2 B-AGM(1)illustration. . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.4.3 Extensionstomorethanonestatevariable . . . . . . . . . . . . . 192 5.5 B-AGMestimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.5.1 Partialestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.5.2 Fullestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.5.3 Alternativeestimationmethods . . . . . . . . . . . . . . . . . . . . 206 5.6 ApproximationstoB-AGMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.6.1 K-AGMsasapproximationstoB-AGMs. . . . . . . . . . . . . . . 206

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