固定收益证券Collaterized-Debt-Obligations课件

Chapter16

CollaterizedDebtObligations

Chapter16

CollaterizedDebtLearningObjectivesAfterreadingthischapter,youwillunderstandwhatismeantbyacollateralizeddebtobligation,collateralizedbondobligation,andcollateralizedloanobligationthestructureofacollateralizeddebtobligationandtheroleofthecollateralmanagerthedifferencebetweenanarbitrageandbalancesheettransactiontheeconomicsunderlyinganarbitragetransactionthemotivationforabalancesheettransactionLearningObjectivesAfterreadiLearningObjectives(continued)Afterreadingthischapter,youwillunderstandthedifferencebetweenacashflowtransactionandamarketvaluetransactionthetypesofrestrictionsimposedonmanagementinacollateralizeddebtobligationthedifferencebetweenacashandsynthetictransactiontheneedforaninterest-rateswapinacashtransactiontheroleofacreditdefaultswapinasynthetictransactionLearningObjectives(continued固定收益证券Collaterized-Debt-Obligations课件固定收益证券Collaterized-Debt-Obligations课件StructureofaCDO(continued)TheproceedstomeettheobligationstotheCDOtranches(interestandprincipalrepayment)cancomefrom:couponinterestpaymentsfromthecollateralassetsmaturingofcollateralassetssaleofcollateralassetsInatypicalstructure,oneormoreofthetrancheshasafloatingrate.Withtheexceptionofdealsbackedbybankloansthatpayafloatingrate,thecollateralmanagerinvestsinfixed-ratebonds.Thiscreatesaproblemasthemanagerpaystrancheinvestorsafloatingratewhileinvestinginassetswithafixedrate.StructureofaCDO(continued)StructureofaCDO(continued)ArbitrageVersusBalanceSheetTransactionsCDOsarecategorizedaseitherarbitragetransactionsorbalancesheettransactions.Thecategorizationdependsonthemotivationofthesponsorofthetransaction.Inanarbitragetransaction,thesponsorseekstoearnthespreadbetweenthehigheryieldreceivedonthecollateralassetsandtheloweryieldpaidtothevarioustranchesinthestructure.Inabalancesheettransaction,thesponsor’smotivationistoremovedebtinstrumentsfromitsbalancesheet.StructureofaCDO(continued)StructureofaCDO

(continued)CashVersusSyntheticStructuresCDOsarealsoclassifiedintermsofcashCDOstructures

andsyntheticCDOstructures.Thelatterinvolvetheuseofcreditderivatives.Attheoutsetofthischapter,wewillfocusoncashCDOstructures.ThelastsectionofthischapterwillcoversyntheticCDOstructures.StructureofaCDO(continued)ArbitrageTransactionsThekeyastowhetheritiseconomicallyfeasibletocreateanarbitrageCDOiswhetherastructurecanofferacompetitivereturnforthesubordinate/equitytranche.TheeconomicsofarbitrageCDOstructuresshowtheneedfortheuseofaninterest-rateswap,andhowthesubordinate/equitytranchewillrealizeareturn.IndeterminingwhetherornottocreateaCDO,dealerswilllooktoseeifthereisapotentialreturnavailabletotheequitytrancheofaminimumamount.Thethresholdreturnisbasedonmarketconditions.ArbitrageTransactionsThekeyArbitrageTransactions(continued)TypesofArbitrageTransactionsArbitragetransactionscanbedividedintotwotypesdependingontheprimarysourceoftheproceedsfromthecollateraltosatisfytheobligationtothetranches.Iftheprimarysourceistheinterestandmaturingprincipalfromthecollateral,thenthetransactionisreferredtoasacashflowtransaction.Ifinsteadtheproceedstomeettheobligationsdependheavilyonthetotalreturngeneratedfromthecollateral(i.e.,interestincome,capitalgain,andmaturingprincipal),thenthetransactionisreferredtoasamarketvaluetransaction.ArbitrageTransactions(continCashFlowTransactionsTypesofArbitrageTransactionsInacashflowtransaction,theobjectiveofthecollateralmanageristogeneratecashflowfortheseniorandmezzaninetrancheswithouttheneedtoactivelytradebonds.Therearethreerelevantperiods.Thefirstistheramp-upperiod.Thisistheperiodthatfollowstheclosingdateofthetransactionwherethecollateralmanagerbeginsinvestingtheproceedsfromthesaleofthedebtobligationsissued.Thisperiodusuallylastsfromonetotwoyears.ThereinvestmentperiodorrevolvingperiodiswhereprincipalproceedsarereinvestedThisperiodusuallylastsforfiveormoreyears.Inthefinalperiod,thecollateralissoldandthedebtholdersarepaidoff.CashFlowTransactionsTypesofCashFlowTransactions(continued)DistributionofIncomeIncomeisderivedfrominterestincomefromthecollateralassetsandcapitalappreciation.Theincomeisusedasfollows.Paymentsarefirstmadetothetrusteeandadministratorsandthentotheseniorcollateralmanager.Oncethesefeesarepaid,thentheseniortranchesarepaidtheirinterest.Atthispoint,beforeanyotherpaymentsaremade,certaintestsmustbepassed.Thesetestsarecalledcoveragetests.Ifthecoveragetestsarepassed,theninterestispaidtothemezzaninetranches.Oncethemezzaninetranchesarepaid,interestispaidtothesubordinate/equitytranche.CashFlowTransactions(continCashFlowTransactions(continued)DistributionofPrincipalCashFlowTheprincipalcashflowisdistributedasfollowsafterthepaymentofthefeestothetrustees,administrators,andseniormanagers.Ifthereisashortfallininterestpaidtotheseniortranches,principalproceedsareusedtomakeuptheshortfall.Afterallthedebtobligationsaresatisfiedinfull,ifpermissible,theequityinvestorsarepaid.Managementispermittedtoshareonsomeproratedbasisoncethetargetreturnisachieved.CashFlowTransactions(continCashFlowTransactions(continued)RestrictionsonManagement:QualityTestsInratingatransaction,theratingagenciesareconcernedwiththediversityoftheassets.Consequently,thereareteststhatrelatetothediversityoftheassetsandthesetestsarecalledqualitytests.Qualitytestsconsider:maturityrestrictionsrestrictionsimposedontheconcentrationofbondsincertaincountriesorgeographicalregionsforcollateralconsistingofemergingmarketbondsAdiversityscoreisameasurethatisconstructedtogaugethediversityofthecollateral’sassets.Thegreaterthescorevalue,thelowerthelikelihoodofdefault.CashFlowTransactions(continCashFlowTransactions(continued)RestrictionsonManagement:QualityTestsOnecandescribethedistributionofthecreditratingsofthecollateralintermsofthepercentageofthecollateral’sassetineachcreditrating.However,thereisaneedtohaveonefigurethatsummarizestheratingdistributiontest.Moody’sandFitchhavedevelopedameasuretosummarizetheratingdistribution.Thisiscommonlyreferredtoastheweighted-averageratingfactor

(WARF)forthecollateral.UnlikeMoody’sandFitch,S&Pusesadifferentsystem.S&Pspecifiesrequiredratingpercentagesthatthecollateralmustmaintain.Specifically,S&Prequiresstrictpercentagelimitsforlowerratedassetsinthecollateral.CashFlowTransactions(continCashFlowTransactions(continued)RestrictionsonManagement:QualityTestsTherearetwotypesofcoverageteststoensurethattheperformanceofthecollateralissufficienttomakepaymentstothevarioustranches.Thesetwotypesarecalledparvaluetestsandinterestcoverageratiotests.Aseparateparvaluetestisusedforeachratedbondissuedinthetransaction.Aparvaluetestspecifiesthattheparvalueofthecollateralbeatleastaspecifiedpercentageabovetheliabilitytothebondholders.Anovercollateralizationtestforaratedbondissuedisameasureofthecushionprovidedbythecollateral’sassetsovertheobligationtothebondholdersintermsofparvalue.CashFlowTransactions(continCashFlowTransactions(continued)RestrictionsonManagement:QualityTestsThepercentageintheparvaluetestiscalledthetrigger,

andthetriggerisdifferentforeachratedbond.Specifically,thetriggerdeclinesastheratingdeclines.Whileparvaluetestsfocusonthemarketvalueofthecollateralrelativetotheparvalueofthebondsissued,interestcoveragetestslookattheabilitytomeetinterestpaymentswhendue.CashFlowTransactions(continMarketValueTransactionsInamarketvaluetransaction,thecashflowgeneratedtopaythebondholdersdependsupontheabilityofthecollateralmanagertomaintainandimprovethemarketvalueofthecollateral.Fundstobeusedforliabilityprincipalpaymentsareobtainedfromliquidatingthecollateral.Liabilityinterestpaymentscanbemadefromcollateralinterestreceipts,aswellascollateralliquidationproceeds.Ratingsarebasedonpricevolatility,liquidity,andmarketvalueofthecollateralassets.Thecollateralmanagerfocusesonmaximizingtotalreturnwhileminimizingvolatility.MarketValueTransactionsInaMarketValueTransactions

(continued)Theorderofpriorityoftheprincipalpaymentsinthecapitalstructureisasfollows.Feesarepaidfirstfortrustees,administrators,andmanagers.Afterthesefeesarepaid,theseniorfacilityclassandtheseniornotesclassarepaid.Thesetwoclassesinthecapitalstructurearetreatedequallyintheirrightstotheirclaimoncashproceedsfromthecollateral.Thesenior-subordinatednoteswouldbepaid,followedbythesubordinatednotes.Allofthisassumesthattheovercollateralizationtestsaresatisfied.Ifnot,theseniornotesarethenpaiddownuntiltheovercollateralizationtestsarebroughtintocompliance.MarketValueTransactions(conMarketValueTransactions

(continued)Whenratingacashflowtransaction,theratingagencieslookattheabilityofthecollateraltogeneratesufficientcurrentcashflowtopayinterestandprincipalonratednotesissuedbytheCDO.Theratingsarebasedontheeffectofcollateraldefaultsandrecoveriesonthereceiptoftimelyinterestandprincipalpaymentsfromthecollateral.Itisthejobofthecollateralmanagertoconcentrateeffortsoncontrollingdefaultsandrecoveries.Iftheovercollateralizationtestsarenotmet,thencashflowisdivertedfromthemezzanineandsubordinatedclassestopaydownseniornotes,orcashflowistrappedinareserveaccount.Failingtheovercollateralizationtestsdoesnotforcesaleofthecollateral.

MarketValueTransactions(conMarketValueTransactions(continued)OvercollateralizationTestsOvercollateralizationtestsinmarketvaluetransactionsarebasedonthemarketvalueofthecollateral,nottheparvalue.Marketvalueovercollateralizationtestsrequirethatthemarketvalueofthecollateralbeadjustedtoobtainanadjustedmarketvalueforthecollateral.Theadvanceratesarethekeyintheovercollateralizationtestsandcriticalinmarketvaluetransactions.Advanceratesaredeterminedbytheratingagenciesbasedonacombinationofthreefactors:pricevolatilitycorrelationamongsecuritiesliquidityMarketValueTransactions(conMarketValueTransactions(continued)OvercollateralizationTestsThereisthenanadvancerateassignedtoeachassettypebasedonthestructureofthetransaction,andthecompositionofthecollateral.Forexample,supposethatastructurehasonlyoneratedtranche.Thismeansthatthereisonlyaseniortrancheandnomezzaninetranche.Consequently,alloftheprotectionfortheseniortranchemustcomefromthecollateral.Thebelowtableshowstheadvanceratesforperforminghigh-yieldbondsratedBassignedbyMoody’stoobtainatargetratingofAaa,Aa3,A3,orBaa3ifthecollateralcontainsoneassettype:

TargetRatingAaa Aa3 A3 Baa320Issuersand5Industries 0.72 0.77 0.80 0.8540Issuersand10Industries 0.74 0.80 0.83 0.86MarketValueTransactions(conMarketValueTransactions

(continued)OvercollateralizationTestsSupposethatthecollateralconsistsofthreeassettypeswiththeassumedadvanceratingsfortheparticularratingsoughtforatranche:Themarketvalueofthecollateralis$100million.Theadjustedmarketvaluethatmustbeusedintheovercollateralizationtestsforthistranchewouldthenbefoundbymultiplyingthemarketvalueofanassettypebytheadvancerateandthensummingoverallassettypes.So,forourhypotheticalcollateral,theadjustedmarketvalueisfoundasfollows:($50M×0.80)+($30M×0.75)+($20Mx0.70)=$76,500,000

AssetTypeMarketValuePerformingHigh-YieldBondsRatedBaa $50million 0.80PerformingHigh-YieldBondsRatedB $30million 0.75PerformingHigh-YieldBondsValuedBelowCaa $20million 0.70AdvanceRateMarketValueTransactions(conSyntheticCDOsCashCDOstructuresaresonamedbecausethecollateralassetsareowned.Inrecentyears,thefastestgrowingsectoroftheCDOmarketisthesyntheticCDOstructure.Thenamefollowsfromthefactthatthecollateralassetsarenotactuallyowned.InasyntheticCDOthecollateralabsorbstheeconomicrisksassociatedwithspecifiedassetsbutdoesnothavelegalownershipofthoseassets.SyntheticCDOsCashCDOstructuSyntheticCDOs(continued)ThecreationofasyntheticCDOstructurerequirestheuseofacreditderivative.Morespecifically,thetypeofcreditderivativeusedisacreditdefaultswap.Acreditdefaultswapallowsmarketparticipantsthatownanassettotransferthecreditriskassociatedwiththatassettoanotherpartywithouttransferringthelegalownershipofthatasset.SyntheticCDOs(continued)TheSyntheticCDOs(continued)Foracreditdefaultswap,thereisacreditprotectionbuyerandacreditprotectionseller.Thecreditprotectionbuyerpaysafee(premium)tothecreditprotectionseller.Ifa“creditevent”occurs,thenthecreditprotectionsellermustmakeapaymenttothecreditprotectionbuyer.Crediteventsonadebtinstrumentmayincludebankruptcy,failuretopaywhendue,downgradingofanissue,debtrepudiation,anddebtrestructuring.SyntheticCDOs(continued)ForSyntheticCDOs(continued)Withbasicinformationaboutcreditdefaultswaps,wecanlookatthebasicstructureofasyntheticCDO.AswithacashCDOstructure,liabilitiesareissued.Theproceedsreceivedfromthetrancheswillbeinvestedbythecollateralmanagerinassetswithlowrisk.Inaddition,thecollateralmanagerwillenterintoacreditdefaultswapwithanotherentityinwhichitwillprovidecreditprotection.Becauseitissellingcreditprotection,thecollateralmanagerwillreceivethecreditdefaultswapfee.SyntheticCDOs(continued)WithSyntheticCDOs(continued)Ontheothersideofthecreditdefaultswapwillbeacreditprotectionbuyerwhowillbepayingthefee.Thisentitywillbeafinancialinstitutionseekingtoshedthecreditriskofsomeofitsassets.Forexample,itcouldbeabankthatisusingthecreditdefaultswapforsomespecificallydefinedloansinthebank’sportfolio.Theseloansarereferredtoasthereferenceassetsinthecreditdefaultswap.SyntheticCDOs(continued)OntSyntheticCDOs(continued)Ifacrediteventdoesnotoccur,thereturnrealizedbythecollateralmanagerthatwillbeavailabletomeetthestructure’sobligationswillbethereturnonthecollateralconsistingoflowriskassetsplusthefeereceivedfromthecreditdefaultswap.Ifthereisadefaultonanyofthereferencedassets,thecollateralmanagermustmakeapaymenttothecounterparty.Thisreducesthereturnavailabletomeetthestructure’sobligations.SyntheticCDOs(continued)IfaSyntheticCDOs(continued)Therearestructuresthatincludeelementsofbothcashflowandsynthetictransactions.Thatis,withtheproceedsobtainedfromthesaleofthebondclasses,thecollateralmanagerusesaportiontopurchasefortheportfoliobondsand/orloans(asinacashflowtransaction)investsthebalanceinhigh-qualitydebtinstruments(asinasynthetictransaction)sellscreditprotectiononselectedreferenceentities(asinasynthetictransaction)SyntheticCDOs(continued)TherAllrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,withoutthepriorwrittenpermissionofthepublisher.PrintedintheUnitedStatesofAmerica.Allrightsreserved.NopartoChapter17

Interest-RateModels

Chapter17

Interest-RateModeLearningObjectivesAfterreadingthischapter,youwillunderstandwhataninterest-ratemodelishowaninterest-ratemodelisrepresentedmathematicallythecharacteristicsofaninterest-ratemodel:drift,volatility,andmeanreversionwhataone-factorinterest-ratemodelisthedifferencebetweenanarbitrage-freemodelandanequilibriummodelthedifferenttypesofarbitrage-freemodelsandwhytheyareusedinpracticethedifferencebetweenanormalmodelandalognormalmodeltheempiricalevidenceoninterestratechangesconsiderationsinselectinganinterestratemodelhowtocalculatehistoricalvolatilityLearningObjectivesAfterreadiMathematicalDescriptionofOne-FactorInterest-RateModelsInterest-ratemodelsmustincorporatestatisticalpropertiesofinterest-ratemovements.ThesepropertiesaredriftvolatilitymeanreversionThecommonlyusedmathematicaltoolfordescribingthemovementofinterestratesthatcanincorporatethesepropertiesisstochasticdifferentialequations(SDEs).Arigoroustreatmentofinterest-ratemodelingrequiresanunderstandingofthisspecializedtopicinmathematics.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Themostcommoninterest-ratemodelusedtodescribethebehaviorofinterestratesassumesthatshort-terminterestratesfollowsomestatisticalprocessandthatotherinterestratesinthetermstructurearerelatedtoshort-termrates.Theshort-terminterestrate(i.e.,shortrate)istheonlyonethatisassumedtodrivetheratesofallothermaturities.Hence,thesemodelsarereferredtoasone-factormodelswherethe“onefactor”istheshortrate.Theotherratesarenotrandomlydeterminedoncetheshortrateisspecified.Usingarbitragearguments,therateforallothermaturitiesisdetermined.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Therearealsomulti-factormodelsthathavebeenproposedintheliterature.Themostcommonmulti-factormodelisatwo-factormodelwherealong-termrateisthesecondfactor.Inpractice,however,one-factormodelsareusedbecauseofthedifficultyofapplyingevenatwo-factormodelaswellasempiricalevidencethatsupportsone-factormodels.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Whilethevalueoftheshortrateatsomefuturetimeisuncertain,thepatternbywhichitchangesovertimecanbeassumed.Instatisticalterminology,thispatternorbehavioriscalledastochasticprocess.Thus,describingthedynamicsoftheshortratemeansspecifyingthestochasticprocessthatdescribesthemovementoftheshortrate.Itisassumedthattheshortrateisacontinuousrandomvariableandthereforethestochasticprocessusedisacontinuous-timestochasticprocess.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Therearedifferenttypesofcontinuous-timestochasticprocessesusedininterest-ratemodeling.Inallofthesemodelsbecausetimeisacontinuousvariable,theletterdisusedtodenotethe“changein”somevariable.Specifically,inthemodelswelet

r=theshortrateandthereforedrdenotesthechangeintheshortratet=timeandthusdtdenotesthechangeintimeorequivalentlythelengthofthetimeinterval(foraverysmallintervaloftime)z=arandomtermanddzdenotesarandomprocessMathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)ABasicContinuous-TimeStochasticProcessAbasiccontinuous-timestochasticprocessfordescribingthedynamicsoftheshortrate(r)isgivenby:dr=bdt+σdz

dr=thechangeintheshortrateb=expecteddirectionofratechangedt=thechangeintimeorequivalentlythelengthofthetimeinterval(foraverysmallintervaloftime)σ=standarddeviationofthechangesintheshortratez=arandomtermanddzdenotesarandomprocessTheexpecteddirectionofthechangeintheshortrate(b)iscalledthedrifttermandσiscalledthevolatilityterm.Thechangeintheshortrate(dr)overthetimeinterval(dt)dependsontheexpecteddirectionofthechangeintheshortrate(b)arandomprocess(dz)thatisaffectedbyvolatilityMathematicalDescriptionofOnMathematicalDescriptionofOne-Factor

Interest-RateModels(continued)ABasicContinuous-TimeStochasticProcessTherandomnatureofthechangeintheshortratecomesfromtherandomprocessdz.Theassumptionsarethattherandomtermzfollowsanormaldistributionwithameanofzeroandastandarddeviationofone(i.e.,isastandardizednormaldistribution)thechangeintheshortrateisproportionaltothevalueoftherandomterm,whichdependsonthestandarddeviationofthechangeintheshortratethechangeintheshortrateforanytwodifferentshortintervalsoftimeisindependent

Theexpectedvalueofthechangeintheshortrateisequaltob,thedriftterm.Inthespecialcasewherethedrifttermiszeroandthevarianceisone,itcanbeshownthatthevarianceofthechangeintheshortrateoversomeintervaloflengthTisequaltoTandthereforethestandarddeviationisthesquarerootofT.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)ItôProcessNoticethatintheequation,dr=bdt+σdz,thatneitherthedriftterm(b)northestandarddeviationofthechangeintheshortrate(σ)dependsoneithertheleveloftheshortrate(r)andtime(t).Thereareeconomicreasonsthatmightsuggestthattheexpecteddirectionoftheratechangewilldependonthelevelofthecurrentshortrate;thesameistrueforσ.Wecanchangethedynamicsofthedrifttermandthedynamicsofthevolatilitytermbyallowingthesetwoparameterstodependontheleveloftheshortrateand/ortime.Wecandenotethatthedrifttermdependsonboththeleveloftheshortrateandtimebyb(r,t);thesameistrueforσ,e.g.,σ(r,t).Thus,wecanwritedr=b(r,t)dt+σ(r,t)dz

Thecontinuous-timestochasticmodelgivenbytheaboveequationiscalledanItoprocess.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheDriftTermInspecifyingthedynamicsofthedriftterm,onecanspecifythatthedrifttermdependsontheleveloftheshortratebyassumingitfollowsameanreversionprocess.Bymeanreversionitismeantthatsomelong-runstablemeanvaluefortheshortrateisassumed.Wedenotethisvalueby.So,ifrisgreaterthan,thedirectionofchangeintheshortratewillmovedowninthedirectionofthelong-runstablevalueandviceversa.Themeanreversionprocessthatspecifiesthedynamicsofthedrifttermis:

b(r,t)=α(r–)whereαiscalledthespeedofadjustmentbecauseitindicatesthespeedatwhichtheshortratewillmoveorconvergetothelong-runstablemeanvalue.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermTherehavebeenseveralformulationsofthedynamicsofthevolatilityterm.Ifvolatilityisnotassumedtodependontime,thenσ(r,t)=σ(r).Ingeneral,thedynamicsofthevolatilitytermcanbespecifiedasfollows:σrγdzwhereγisequaltotheconstantelasticityofvariance.Theaboveequationiscalledtheconstantelasticityofvariancemodel(CEVmodel).TheCEVmodelallowsustodistinguishbetweenthedifferentspecificationsofthedynamicsofthevolatilitytermforthevariousinterest-ratemodelssuggestedbyresearchers.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermFortheVasicekinterestratemodel,welookatthecaseforγ=0.Substitutingzeroforγintotheequationσrγdz,wegetthefollowingtheCEVmodelidentifiedbyVasicekwhofirstproposedit:γ=0:σ(r,t)=σIntheVasicekspecificationoftheCEVmodel,volatilityisindependentoftheleveloftheshortrateasintheequationofdr=bdt+σdz(wherebisthedriftterm)andisreferredtoasthenormalmodel.Inthenormalmodel,itispossiblefornegativeinterestratestobegenerated.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermFortheDothaninterestratemodel,welookatthecaseforγ=1.Substitutingoneforγintotheequationσrγdz,wegetthefollowingtheCEVmodelspecifiedbyDothanwhofirstproposedit:γ=1:σ(r,t)=σrIntheDothanspecificationoftheCEVmodel,volatilityisproportionaltotheshortrate.Thismodelisreferredtoastheproportionalvolatilitymodel.MathematicalDescriptionofOnMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermFortheCox-Ingersoll-Ross(CIR)interestratemodel,welookatthecaseforγ=½

.Substitutingoneforγintotheequationσrγdz,wegetthefollowingtheCEVmodelproposedbyCIR:γ=½:σ(r,t)=TheCIRspecification,referredtoasthesquare-rootmodel,makesthevolatilityproportionaltothesquarerateoftheshortrate.Negativeinterestratesarenotpossibleinthissquare-rootmodel.MathematicalDescriptionofOnArbitrage-FreeVersusEquilibriumModelsArbitrage-FreeModelsInarbitrage-freemodels,alsoreferredtoasno-arbitragemodels,theanalysisbeginswiththeobservedmarketpriceofasetoffinancialinstruments.Thefinancialinstrumentscanincludecashmarketinstrumentsandinterest-ratederivati