固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件

FIXED-INCOMESECURITIESLecture9OptionsonBondsandBondswithEmbeddedOptionsFIXED-INCOMESECURITIES1固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件2固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件3ValueofThree-PeriodOption-FreeBond

C=9,F=100

ValueofThree-PeriodOption-F4CallableBondsandPutableBonds

BondwithEmbeddedOptionsCallablebonds–Issuermayrepurchaseatapre-specifiedcallprice–TypicallycalledifinterestratesfallAcallablebondhastwodisadvantagesforaninvestor–Ifitiseffectivelycalled,theinvestorwillhavetoinvestinanotherbondyieldingalowerrate–Acallablebondhastheunpleasantpropertyforaninvestortoappreciatelessthananormalsimilarbondwheninterestratesfall–Therefore,aninvestorwillbewillingtobuysuchabondatalowerpricethanacomparableoption-freebondExamples–TheUKTreasurybondwithcoupon5.5%andmaturitydate09/10/2012canbecalledinfullorpartfrom09/10/2008onatapriceofpounds100–TheUSTreasurybondwithcoupon7.625%andmaturitydate02/15/2007canbecalledoncoupondatesonly,atapriceof$100,from02/15/2002on–SuchabondissaidtobediscretelycallableCallableBondsandPutableBon5CallableandPutableBonds

InstitutionalAspectsPutablebondholdermayretireatapre-specifiedpriceAputablebondallowsitsholdertosellthebondatparvaluepriortomaturityincaseinterestratesexceedthecouponrateoftheissueSo,hewillhavetheopportunitytobuyanewbondatahighercouponrateTheissuerofthisbondwillhavetoissueanotherbondatahighercouponrateiftheputoptionisexercisedHenceaputablebondtradesatahigherpricethanacomparableoption-freebondCallableandPutableBo6CallableandPutableBonds

Yield-to-WorstYield-to-callYear54.54%Year64.61%Year74.66%Year84.69%Year94.72%Yield-to-worstyear104.74%LetusconsiderabondwithanembeddedcalloptiontradingoveritsparvalueThisbondcanberedeemedbyitsissuerpriortomaturity,fromitsfirstcalldateon–Onecancomputeayield-to-callonallpossiblecalldates–Theyield-to-worstisthelowestoftheyield-to-maturityandallyields-to-callExample–10-yearbondbearinganinterestcouponof5%,discretelycallableafter5yearsandtradingat102–Thereare5possiblecalldatesbeforematurity–Yield-to-worstis4.54%CallableandPutableBonds7CallableBonds

ValuationinaBinomialModel

thevalueofthecallablebondisdeterminedbyselectingtheminimumoftheotherwisenoncallablebondorthecallprice,andthenrollingthecallablebondvaluetothecurrentperiod.Recursiveprocedure–Pricecash-flowtobediscountedonperiodn-1istheminimumvalueofthepricecomputedonperiodnandcallpriceonperiodn–AndsoonuntilwegetthepricePofthecallablebondCallableBonds

Valuationina8固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件9固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件10ValueofPutablebondValueofPutablebond11固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件12Analternativebutequivalentapproachistocalculatetheweightedaveragevalueofeachpossiblepathsdefinedbythebinomialprocess.Thisvalueisknownasthetheoreticalvalue.

Thet-periodspotrateisequaltothegeometricaverageofthecurrentandexpectone-periodspotrates.

AlternativeBinomialValuationApproach

Analternativebutequivalent13consideragainthethree-period,9%option-freebondvaluedwithatwo-periodinterestratetree

consideragainthethree-perio14固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件15固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件16固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件17CallableandPutableBonds

MonteCarloApproachStep1:generatealargenumberofshort-terminterestratepathsStep2:alongeachinterestratepath,thepricePofthebondwithembeddedoptionisrecursivelydeterminedThepriceofthebondiscomputedastheaverageofitspricesalongallinterestratepathsCallableandPutableBonds

Mon18CallableandPutableBonds

MonteCarloApproach-ExamplePriceacallablebondwithannualcoupon4.57%,maturity10years,redemptionvalue100andcallableat100after5yearsPricesofthebondundereachscenario•PriceofthebondisaverageoverallpathsP=1/6(100.43+100.55+99.9+99.76+99.68+100.55)=100.14•TheMonteCarlopricingmethodologycanalsobeappliedtothevaluationofallkindsofinterestratesderivativesCallableandPutableBonds

Mon19OptionsonBonds

TerminologyAnoptionisacontractinwhichtheseller(writer)grantsthebuyertherighttopurchasefrom,orsellto,theselleranunderlyingasset(hereabond)ataspecifiedpricewithinaspecifiedperiodoftimeThesellergrantsthisrighttothebuyerinexchangeforacertainsumofmoneycalledtheoptionpriceoroptionpremiumThepriceatwhichtheinstrumentmaybeboughtorsoldiscalledtheexerciseorstrikepriceThedateafterwhichanoptionisvoidiscalledtheexpirationdate–AnAmericanoptionmaybeexercisedanytimeuptoandincludingtheexpirationdate–AEuropeanoptionmaybeexercisedonlyontheexpirationdateOptionsonBonds

TerminologyAn20OptionsonBonds

FactorsthatInfluenceOptionPrices

Currentpriceofunderlyingsecurity–Asthepriceoftheunderlyingbondincreases,thevalueofacalloptionrisesandthevalueofaputoptionfalls•Strikeprice–Call(put)optionsbecomemore(less)valuableastheexercisepricedecreasesTimetoexpiration–ForAmericanoptions,thelongerthetimetoexpiration,thehighertheoptionpricebecauseallexerciseopportunitiesopentotheholderoftheshort-lifeoptionarealsoopentotheholderofthelong-lifeoption•Short-termrisk-freeinterestrate–Priceofcalloptiononbondincreasesandpriceofputoptiononbonddecreasesasshort-terminterestraterises(throughimpactonbondprice)•Expectedvolatilityofyields(orprices)–Astheexpectedvolatilityofyieldsoverthelifeoftheoptionincreases,thepriceoftheoptionwillalsoincreaseOptionsonBonds

Factorsthat21OptionsonBonds

PricingOptionsonlong-termbonds–Interestpaymentsaresimilartodividends–Otherwise,long-termbondsarelikeoptionsonstock:–WecanuseBlack-Scholesasinoptionsondividend-payingequity•Optionsonshort-termbonds–Problem:theyarenotlikeastockbecausetheyquicklyconvergetopar–WecannotdirectlyapplyBlack-Scholes•Othershortcomingsofstandardoptionpricingmodels–Assumptionofaconstantshort-termrateisinappropriateforbondoptions–Assumptionofaconstantvolatilityisalsoinappropriate:asabondmovesclosertomaturity,itspricevolatilitydeclineOptionsonBonds

PricingOption22OptionsonBonds

PricingAsolutiontoavoidtheproblemistoconsideraninterestratemodel,–Thefollowingfigureshowsatreeforthe1-yearrateofinterest(calibratedtothecurrentTS)–Thefigurealsoshowsthevaluesforadiscountbond(par=100)ateachnodeinthetreeOptionsonBonds

PricingAsolu23OptionsonBonds

PricingConsidera2-yearEuropeancallonthis3-yearbondstruckat93.5Startbycomputingthevalueattheendofthetree–Ifbytheendofthe2ndyeartheshort-termratehasrisento7%andthebondistradingat93,theoptionwillexpireworthless–Ifthebondistradingat94(correspondingtoashort-termrateof6%)thecalloptionisworth0.5–Ifthebondistradingat95(short-termrate=5%),thecallisworth1.5WorkingourwaybackwardthetreeOptionsonBonds

PricingConsid24OptionsonBonds

Put-CallParityAssumptionnocouponpaymentsandnoprematureexerciseConsideraportfoliowherewepurchaseonezerocouponbond,oneputEuropeanoption,andsell(write)oneEuropeancalloption(sametimetomaturityTandthesamestrikepriceX)PayoffatdateTOptionsonBonds

Put-CallPari25OptionsonBonds

Put-CallParity–Con’tNomatterwhatstateoftheworldobtainsattheexpirationdate,theportfoliowillbeworthXThus,thepayofffromtheportfolioisrisk-free,andwecandiscountitsvalueattherisk-freeraterWeobtainthecall-putrelationship•ForcouponbondsOptionsonBonds

Put-CallPari26ConvertibleBonds

DefinitionConvertiblesecuritiesareusuallyeitherconvertiblebondsorconvertiblepreferredshareswhicharemostoftenexchangeableintothecommonstockofthecompanyissuingtheconvertiblesecurityBeingdebtorpreferredinstruments,theyhaveanadvantagetothecommonstockincaseofdistressorbankruptcyConvertiblebondsoffertheinvestorthesafetyofafixedincomeinstrumentcoupledwithparticipationintheupsideoftheequitymarketsEssentially,convertiblebondsarebondsthat,attheholder'soption,areconvertibleintoaspecifiednumberofsharesConvertibleBonds

DefinitionCo27ConvertibleBonds

TerminologyConvertiblebonds–Bondholderhasarighttoconvertbondforpre-specifiednumberofshareofcommonstockTerminology–Convertiblepriceisthepriceoftheconvertiblebond–Bondfloororinvestmentvalueisthepriceofthebondifthereisnoconversionoption–Conversionratioisthenumberofsharesthatisexchangedforabond–Conversionvalue=currentsharepricexconversionratio–Conversionpremium=(convertibleprice–conversionvalue)/conversionvalue

ConvertibleBonds

TerminologyC28ConvertibleBonds

ExamplesExample1:–Currentbondprice=$930–Conversionratio:1bond=30sharescommon–Currentstockprice=$25/share–MarketConversionValue=(30shares)x(25)=$750–ConversionPremium=(930–750)/750=180/750=24%•Example2:AXAConvertibleBond–AXAhasissuedinthe€zoneaconvertiblebondpayinga2.5%couponrateandmaturingon01/01/2014;theconversionratiois4.04–On12/13/2001,thecurrentsharepricewas€24.12andthebid-askconvertiblepricewas156.5971/157.5971–Theconversionvaluewasequalto€97.44=4.04x24.12–Theconversionpremiumcalculatedwiththeaskprice157.5971was61.73%=(157.5791-97.44)/97.44–Theconversionofthebondinto4.04sharescanbeexecutedonanydatebeforethematuritydateConvertibleBonds

ExamplesExam29ConvertibleBonds

UsesFortheissuer–Issuingconvertiblebondsenablesafirmtoobtainbetterfinancialconditions–Couponrateofsuchabondisalwayslowertothatofabulletbondwiththesamecharacteristicsintermsofmaturityandcouponfrequency–Thiscomesdirectlyfromtheconversionadvantagewhichisattachedtothisproduct–BesidestheexchangeofbondsforsharesdiminishestheliabilitiesofthefirmissuerandincreasesinthesametimeitsequitysothatitsdebtcapacityisimprovedFortheconvertiblebondholder–Theconvertiblebondisadefensivesecurity,verysensitivetoariseinthesharepriceandprotectivewhenthesharepricedecreases–Ifthesharepriceincreases,theconvertiblepricewillalsoincrease–Whensharepricedecreases,priceofconvertiblenevergetsbelowthebondfloor,i.e.,thepriceofanotherwiseidenticalbulletbondwithnoconversionoptionConvertibleBonds

UsesForthe30ConvertibleBonds

DeterminantsofConvertibleBondPricesConvertiblebondissimilartoanormalcouponbondplusacalloptionontheunderlyingstock

–Withanimportantdifference:theeffectivestrikepriceofthecalloptionwillvarywiththepriceofthebondConvertiblesecuritiesarepricedasafunctionof

–Thepriceoftheunderlyingstock–Expectedfuturevolatilityofequityreturns–Riskfreeinterestrates–Callprovisions–Supplyanddemandforspecificissues–Issue-specificcorporate/Treasuryyieldspread–Expectedvolatilityofinterestratesandspreads•Thus,thereislargeroomforrelativemis-valuationsConvertibleBonds

Determinants31ConvertibleBonds

ConvertibleBondPriceasaFunctionofStockPriceConvertibleBonds

Convertible32ConvertibleBonds

ConvertibleBondPricingModelApopularmethodforpricingconvertiblebondsisthecomponentmodel–Theconvertiblebondisdividedintoastraightbondcomponentandacalloptionontheconversionprice,withstrikepriceequaltothevalueofthestraightbondcomponent–Thefairvalueofthetwocomponentscanbecalculatedwithstandardformulas,suchasthefamousBlack-Scholesvaluationformula.•Thispricingapproach,however,hasseveraldrawbacks–First,separatingtheconvertibleintoabondcomponentandanoptioncomponentreliesonrestrictiveassumptions,suchastheabsenceofembeddedoptions(callabilityandputability,forinstance,areconvertiblebondfeaturesthatcannotbeconsideredintheaboveseparation)–Second,convertiblebondscontainanoptioncomponentwithastochasticstrikepriceequaltothebondpriceConvertibleBonds

Convertible33ConvertibleBonds

ConvertibleBondPricingModelsTheoreticalresearchonconvertiblebondpricingwasinitiatedbyIngersoll(1977)andBrennanandSchwartz(1977),whobothappliedthecontingentclaimsapproachtothevaluationofconvertiblebondsIntheirvaluationmodels,theconvertiblebondpricedependsonthefirmvalueastheunderlyingvariableBrennanandSchwartz(1980)extendtheirmodelbyincludingstochasticinterestrates.ThesemodelsrelyheavilyonthetheoryofstochasticprocessesandrequirearelativelyhighlevelofmathematicalsophisticationConvertibleBonds

Convertible34ConvertibleBonds

BinomialModelThepriceofthestockonlycangouptoagivenvalueordowntoagivenvalueBesides,thereisabond(bankaccount)thatwillpayinterestofrConvertibleBonds

BinomialMod35ConvertibleBonds

BinomialModelWeassumeu(up)>d(down)ForBlackandScholeswewillneedd=1/uForconsistencywealsoneedu>(1+r)>dExample:u=1.25;d=0.80;r=10%ConvertibleBonds

BinomialMod36ConvertibleBonds

BinomialModelBasicmodelthatdescribesasimpleworld.Asthenumberofstepsincreases,itbecomesmorerealisticWewillpriceandhedgeanoption:itappliestoanyotherderivativesecurityKey:wehavethesamenumberofstatesandsecurities(completemarkets)BasisforarbitragepricingConvertibleBonds

BinomialMod37ConvertibleBonds

BinomialModelIntroduceanEuropeancalloption:K=110ItmaturesattheendoftheperiodConvertibleBonds

BinomialMod38ConvertibleBonds

BinomialModelWecanreplicatetheoptionwiththestockandthebondConstructaportfoliothatpaysCuinstateuandCdinstatedThepriceofthatportfoliohastobethesameasthepriceoftheoptionOtherwisetherewillbeanarbitrageopportunityConvertibleBonds

BinomialMod39ConvertibleBonds

BinomialModelWebuysharesandinvestBinthebankTheycanbepositive(buyordeposit)ornegative(shortsellorborrow)Wewantthen,Withsolution,ConvertibleBonds

BinomialMod40ConvertibleBonds

BinomialModelInourexample,wegetforstock:And,forbonds:Thecostoftheportfoliois,ConvertibleBonds

BinomialMod41ConvertibleBonds

BinomialModelThepriceoftheEuropeancallmustbe9.09.Otherwise,thereisanarbitrageopportunity.Ifthepriceislowerthan9.09wewouldbuythecallandshortselltheportfolioIfhigher,theoppositeWehavecomputedthepriceandthehedgesimultaneously:

–Wecanconstructacallbybuyingthestockandborrowing–Shortcall:theoppositeConvertibleBonds

BinomialMod42ConvertibleBonds

BinomialModelRememberthat•And•Substituting,ConvertibleBonds

BinomialMod43ConvertibleBonds

BinomialModelAftersomealgebra,•Observethecoefficients,•Positive•Smallerthanone•AdduptooneLikeaprobability.ConvertibleBonds

BinomialMod44ConvertibleBonds

BinomialModelRewrite•Where•Thiswouldbethepricingof:–Ariskneutralinvestor–Withsubjectiveprobabilitiespand(1-p)ConvertibleBonds

BinomialMod45ConvertibleBonds

BinomialModelSupposethefollowingeconomy,•WeintroduceanEuropeancallwithstrikepriceKthatmaturesinthesecondperiodConvertibleBonds

BinomialMod46ConvertibleBonds

BinomialModelThepriceoftheoptionwillbe:•Thereare“twopaths”thatleadtotheintermediatestate(thatexplainsthe“2”)ConvertibleBonds

BinomialMod47ConvertibleBonds

VolatilityintheBinomialModelConvertibleBonds

Volatilityi48ConvertibleBond

ValuationMethodologyGiventhataconvertiblebondisnothingbutanoptionontheunderlyingstock,weexpecttobeabletousethebinomialmodeltopriceitAteachnode,wetest–a.whetherconversionisoptimal–b.whetherthepositionoftheissuercanbeimprovedbycallingthebonds•Itisadynamicprocedure:max(min(Q1,Q2),Q3)),where–Q1=valuegivenbytherollback(neitherconvertednorcalledback)–Q2=callprice–Q3=valueofstocksifconversiontakesplaceConvertibleBond

ValuationMet49ConvertibleBond

ExampleExample

–Weassumethattheunderlyingstockpricetradesat$50.00witha30%annualvolatility–Weconsideraconvertiblebondwitha9monthsmaturity,aconversionratioof20–Theconvertiblebondhasa$1,000.00facevalue,a4%annualcoupon–Wefurtherassumethattherisk-freerateisa(continuouslycompounded)10%,whiletheyieldtomaturityonstraightbondsissuedbythesamecompanyisa(continuouslycompounded)15%–Wealsoassumethatthecallpriceis$1,100.00–Usea3periodsbinomialmodel(t/n=3months,or¼year)ConvertibleBond

ExampleExampl50ConvertibleBond

ExampleWehave•Actually(continuouslycompoundedrate)ConvertibleBond

ExampleWehav51ConvertibleBond

ExampleConvertibleBond

Example52ConvertibleBond

ExampleAtnodeG,thebondholderoptimallychoosetoconvertsincewhatisobtainedunderconversion($1,568.31),ishigherthanthepayoffundertheassumptionofnoconversion($1,040.00)ThesameappliestonodeHOntheotherhand,atnodesIandJ,thevalueundertheassumptionofconversionislowerthanifthebondisnotconvertedtoequity–Therefore,bondholdersoptimallychoosenottoconvert,andthepayoffissimplythenominalvalueofthebond,plustheinterestpayments,thatis$1,040.00ConvertibleBond

ExampleAtnod53ConvertibleBond

ExampleWorkingourwaybackwardthetree,weobtainatnodeDthevalueoftheconvertiblebondasthediscountedexpectedvalue,usingrisk-neutralprobabilitiesofthepayoffsatnodesGandH•AtnodeF,thesameprincipleapplies,exceptthatitanberegardedasastandardbond•Wethereforeusetherateofreturnonanonconvertiblebondissuedbythesamecompany,15%ConvertibleBond

ExampleWorkin54ConvertibleBond

ExampleAtnodeE,thesituationismoreinterestingbecausetheconvertiblebondwillendupasastockincaseofanupmove(conversion),andasabondincaseofadownmove(noconversion)Asanapproximateruleofthumb,onemayuseaweightedaverageoftheriskfreeandriskyinterestrateinthecomputation,wheretheweightingisperformedaccordingtothe(risk-neutral)probabilityofanupversusadownmove•ThenthevalueiscomputedasConvertibleBond

ExampleAtnod55ConvertibleBond

ExampleNotethatatnodeD,callingorconvertingisnotrelevantbecauseitdoesnotchangethebondvaluesincethebondisalreadyessentiallyequityAtnodeB,itcanbeshownthattheissuerfindsitoptimaltocallthebondIfthebondisindeedcalledbytheissuer,bondholdersareleftwiththechoicebetweennotconvertingandgettingthecallprice($1,100),orconvertingandgetting$20x58.09=1,161.8$,whichiswhattheyoptimallychooseThisislessthan$1,191.13,thevalueoftheconvertiblebondifitwerenotcalled,andthisispreciselywhyitiscalledbytheissuerEventually,thevalueatnodeA,i.e.,thepresentfairvalueoftheconvertiblebond,iscomputedas$1,115.41ConvertibleBond

ExampleNotet56ConvertibleBonds

ConvertibleArbitrageConvertiblearbitragestrategiesattempttoexploitanomaliesinpricesofcorporatesecuritiesthatareconvertibleintocommonstocksRoughlyspeaking,iftheissuerdoeswell,theconvertiblebondbehaveslikeastock,iftheissuerdoespoorly,theconvertiblebondbehaveslikedistresseddebtConvertiblebondstendstobeunder-pricedbecauseofmarketsegmentation:investorsdiscountsecuritiesthatarelikelytochangetypesConvertiblearbitragehedgefundmanagerstypicallybuy(orsometimessell)thesesecuritiesandthenhedgepartoralloftheassociatedrisksbyshortingthestockConvertibleBonds

Convertible57ConvertibleBonds

MechanismInatypicalconvertiblebondarbitrageposition,thehedgefundisnotonlylongtheconvertiblebondposition,butalsoshortanappropriateamountoftheunderlyingcommonstockThenumberofsharesshortedbythehedgefundmanagerisdesignedtomatchoroffsetthesensitivityoftheconvertiblebondtocommonstockpricechanges–Asthestockpricedecreases,theamountlostonthelongconvertiblepositioniscounteredbytheamountgainedontheshortstockposition–Asthestockpriceincreases,theamountgainedonthelongconvertiblepositioniscounteredbytheamountlostontheshortstockpositionThisisknownasdeltahedgingOver-hedgingissometimesappropriatewhenthereisconcernaboutdefault,astheexcessshortpositionmaypartiallyhedgeagainstareductionincreditqualityConvertibleBonds

MechanismIn58FIXED-INCOMESECURITIESLecture9OptionsonBondsandBondswithEmbeddedOptionsFIXED-INCOMESECURITIES59固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件60固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件61ValueofThree-PeriodOption-FreeBond

C=9,F=100

ValueofThree-PeriodOption-F62CallableBondsandPutableBonds

BondwithEmbeddedOptionsCallablebonds–Issuermayrepurchaseatapre-specifiedcallprice–TypicallycalledifinterestratesfallAcallablebondhastwodisadvantagesforaninvestor–Ifitiseffectivelycalled,theinvestorwillhavetoinvestinanotherbondyieldingalowerrate–Acallablebondhastheunpleasantpropertyforaninvestortoappreciatelessthananormalsimilarbondwheninterestratesfall–Therefore,aninvestorwillbewillingtobuysuchabondatalowerpricethanacomparableoption-freebondExamples–TheUKTreasurybondwithcoupon5.5%andmaturitydate09/10/2012canbecalledinfullorpartfrom09/10/2008onatapriceofpounds100–TheUSTreasurybondwithcoupon7.625%andmaturitydate02/15/2007canbecalledoncoupondatesonly,atapriceof$100,from02/15/2002on–SuchabondissaidtobediscretelycallableCallableBondsandPutableBon63CallableandPutableBonds

InstitutionalAspectsPutablebondholdermayretireatapre-specifiedpriceAputablebondallowsitsholdertosellthebondatparvaluepriortomaturityincaseinterestratesexceedthecouponrateoftheissueSo,hewillhavetheopportunitytobuyanewbondatahighercouponrateTheissuerofthisbondwillhavetoissueanotherbondatahighercouponrateiftheputoptionisexercisedHenceaputablebondtradesatahigherpricethanacomparableoption-freebondCallableandPutableBo64CallableandPutableBonds

Yield-to-WorstYield-to-callYear54.54%Year64.61%Year74.66%Year84.69%Year94.72%Yield-to-worstyear104.74%LetusconsiderabondwithanembeddedcalloptiontradingoveritsparvalueThisbondcanberedeemedbyitsissuerpriortomaturity,fromitsfirstcalldateon–Onecancomputeayield-to-callonallpossiblecalldates–Theyield-to-worstisthelowestoftheyield-to-maturityandallyields-to-callExample–10-yearbondbearinganinterestcouponof5%,discretelycallableafter5yearsandtradingat102–Thereare5possiblecalldatesbeforematurity–Yield-to-worstis4.54%CallableandPutableBonds65CallableBonds

ValuationinaBinomialModel

thevalueofthecallablebondisdeterminedbyselectingtheminimumoftheotherwisenoncallablebondorthecallprice,andthenrollingthecallablebondvaluetothecurrentperiod.Recursiveprocedure–Pricecash-flowtobediscountedonperiodn-1istheminimumvalueofthepricecomputedonperiodnandcallpriceonperiodn–AndsoonuntilwegetthepricePofthecallablebondCallableBonds

Valuationina66固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件67固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件68ValueofPutablebondValueofPutablebond69固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件70Analternativebutequivalentapproachistocalculatetheweightedaveragevalueofeachpossiblepathsdefinedbythebinomialprocess.Thisvalueisknownasthetheoreticalvalue.

Thet-periodspotrateisequaltothegeometricaverageofthecurrentandexpectone-periodspotrates.

AlternativeBinomialValuationApproach

Analternativebutequivalent71consideragainthethree-period,9%option-freebondvaluedwithatwo-periodinterestratetree

consideragainthethree-perio72固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件73固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件74固定收益证券chapter09-Options-on-Bonds-and-Bonds-with-Em课件75CallableandPutableBonds

MonteCarloApproachStep1:generatealargenumberofshort-terminterestratepathsStep2:alongeachinterestratepath,thepricePofthebondwithembeddedoptionisrecursivelydeterminedThepriceofthebondiscomputedastheaverageofitspricesalongallinterestratepathsCallableandPutableBonds

Mon76CallableandPutableBonds

MonteCarloApproach-ExamplePriceacallablebondwithannualcoupon4.57%,maturity10years,redemptionvalue100andcallableat100after5yearsPricesofthebondundereachscenario•PriceofthebondisaverageoverallpathsP=1/6(100.43+100.55+99.9+99.76+99.68+100.55)=100.14•TheMonteCarlopricingmethodologycanalso