固定收益证券chapterPPT课件.ppt

FIXED INCOMESECURITIES Lecture9OptionsonBondsandBondswithEmbeddedOptions 1 2 3 ValueofThree PeriodOption FreeBond C 9 F 100 4 CallableBondsandPutableBondsBondwithEmbeddedOptions Callablebonds 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 Suchabondissaidtobediscretelycallable 5 CallableandPutableBondsInstitutionalAspects Putablebondholdermayretireatapre specifiedpriceAputablebondallowsitsholdertosellthebondatparvaluepriortomaturityincaseinterestratesexceedthecouponrateoftheissueSo hewillhavetheopportunitytobuyanewbondatahighercouponrateTheissuerofthisbondwillhavetoissueanotherbondatahighercouponrateiftheputoptionisexercisedHenceaputablebondtradesatahigherpricethanacomparableoption freebond 6 CallableandPutableBondsYield to Worst LetusconsiderabondwithanembeddedcalloptiontradingoveritsparvalueThisbondcanberedeemedbyitsissuerpriortomaturity fromitsfirstcalldateon Onecancomputeayield to callonallpossiblecalldates Theyield to worstisthelowestoftheyield to maturityandallyields to callExample 10 yearbondbearinganinterestcouponof5 discretelycallableafter5yearsandtradingat102 Thereare5possiblecalldatesbeforematurity Yield to worstis4 54 7 CallableBondsValuationinaBinomialModel thevalueofthecallablebondisdeterminedbyselectingtheminimumoftheotherwisenoncallablebondorthecallprice andthenrollingthecallablebondvaluetothecurrentperiod Recursiveprocedure Pricecash flowtobediscountedonperiodn 1istheminimumvalueofthepricecomputedonperiodnandcallpriceonperiodn AndsoonuntilwegetthepricePofthecallablebond 8 9 10 ValueofPutablebond 11 12 Analternativebutequivalentapproachistocalculatetheweightedaveragevalueofeachpossiblepathsdefinedbythebinomialprocess Thisvalueisknownasthetheoreticalvalue Thet periodspotrateisequaltothegeometricaverageofthecurrentandexpectone periodspotrates AlternativeBinomialValuationApproach 13 consideragainthethree period 9 option freebondvaluedwithatwo periodinterestratetree 14 15 16 17 CallableandPutableBondsMonteCarloApproach Step1 generatealargenumberofshort terminterestratepathsStep2 alongeachinterestratepath thepricePofthebondwithembeddedoptionisrecursivelydeterminedThepriceofthebondiscomputedastheaverageofitspricesalongallinterestratepaths 18 CallableandPutableBondsMonteCarloApproach Example Priceacallablebondwithannualcoupon4 57 maturity10years redemptionvalue100andcallableat100after5yearsPricesofthebondundereachscenario PriceofthebondisaverageoverallpathsP 1 6 100 43 100 55 99 9 99 76 99 68 100 55 100 14 TheMonteCarlopricingmethodologycanalsobeappliedtothevaluationofallkindsofinterestratesderivatives 19 OptionsonBondsTerminology Anoptionisacontractinwhichtheseller writer grantsthebuyertherighttopurchasefrom orsellto theselleranunderlyingasset hereabond ataspecifiedpricewithinaspecifiedperiodoftimeThesellergrantsthisrighttothebuyerinexchangeforacertainsumofmoneycalledtheoptionpriceoroptionpremiumThepriceatwhichtheinstrumentmaybeboughtorsoldiscalledtheexerciseorstrikepriceThedateafterwhichanoptionisvoidiscalledtheexpirationdate AnAmericanoptionmaybeexercisedanytimeuptoandincludingtheexpirationdate AEuropeanoptionmaybeexercisedonlyontheexpirationdate 20 OptionsonBondsFactorsthatInfluenceOptionPrices Currentpriceofunderlyingsecurity Asthepriceoftheunderlyingbondincreases thevalueofacalloptionrisesandthevalueofaputoptionfalls Strikeprice Call put optionsbecomemore less valuableastheexercisepricedecreasesTimetoexpiration ForAmericanoptions thelongerthetimetoexpiration thehighertheoptionpricebecauseallexerciseopportunitiesopentotheholderoftheshort lifeoptionarealsoopentotheholderofthelong lifeoption Short termrisk freeinterestrate Priceofcalloptiononbondincreasesandpriceofputoptiononbonddecreasesasshort terminterestraterises throughimpactonbondprice Expectedvolatilityofyields orprices Astheexpectedvolatilityofyieldsoverthelifeoftheoptionincreases thepriceoftheoptionwillalsoincrease 21 OptionsonBondsPricing Optionsonlong termbonds Interestpaymentsaresimilartodividends Otherwise long termbondsarelikeoptionsonstock WecanuseBlack Scholesasinoptionsondividend payingequity Optionsonshort termbonds Problem theyarenotlikeastockbecausetheyquicklyconvergetopar WecannotdirectlyapplyBlack Scholes Othershortcomingsofstandardoptionpricingmodels Assumptionofaconstantshort termrateisinappropriateforbondoptions Assumptionofaconstantvolatilityisalsoinappropriate asabondmovesclosertomaturity itspricevolatilitydecline 22 OptionsonBondsPricing Asolutiontoavoidtheproblemistoconsideraninterestratemodel Thefollowingfigureshowsatreeforthe1 yearrateofinterest calibratedtothecurrentTS Thefigurealsoshowsthevaluesforadiscountbond par 100 ateachnodeinthetree 23 OptionsonBondsPricing Considera2 yearEuropeancallonthis3 yearbondstruckat93 5Startbycomputingthevalueattheendofthetree Ifbytheendofthe2ndyeartheshort termratehasrisento7 andthebondistradingat93 theoptionwillexpireworthless Ifthebondistradingat94 correspondingtoashort termrateof6 thecalloptionisworth0 5 Ifthebondistradingat95 short termrate 5 thecallisworth1 5Workingourwaybackwardthetree 24 OptionsonBondsPut CallParity AssumptionnocouponpaymentsandnoprematureexerciseConsideraportfoliowherewepurchaseonezerocouponbond oneputEuropeanoption andsell write oneEuropeancalloption sametimetomaturityTandthesamestrikepriceX PayoffatdateT 25 OptionsonBondsPut CallParity Con t Nomatterwhatstateoftheworldobtainsattheexpirationdate theportfoliowillbeworthXThus thepayofffromtheportfolioisrisk free andwecandiscountitsvalueattherisk freeraterWeobtainthecall putrelationship Forcouponbonds 26 ConvertibleBondsDefinition ConvertiblesecuritiesareusuallyeitherconvertiblebondsorconvertiblepreferredshareswhicharemostoftenexchangeableintothecommonstockofthecompanyissuingtheconvertiblesecurityBeingdebtorpreferredinstruments theyhaveanadvantagetothecommonstockincaseofdistressorbankruptcyConvertiblebondsoffertheinvestorthesafetyofafixedincomeinstrumentcoupledwithparticipationintheupsideoftheequitymarketsEssentially convertiblebondsarebondsthat attheholder soption areconvertibleintoaspecifiednumberofshares 27 ConvertibleBondsTerminology Convertiblebonds Bondholderhasarighttoconvertbondforpre specifiednumberofshareofcommonstockTerminology Convertiblepriceisthepriceoftheconvertiblebond Bondfloororinvestmentvalueisthepriceofthebondifthereisnoconversionoption Conversionratioisthenumberofsharesthatisexchangedforabond Conversionvalue currentsharepricexconversionratio Conversionpremium convertibleprice conversionvalue conversionvalue 28 ConvertibleBondsExamples Example1 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 04sharescanbeexecutedonanydatebeforethematuritydate 29 ConvertibleBondsUses Fortheissuer Issuingconvertiblebondsenablesafirmtoobtainbetterfinancialconditions Couponrateofsuchabondisalwayslowertothatofabulletbondwiththesamecharacteristicsintermsofmaturityandcouponfrequency Thiscomesdirectlyfromtheconversionadvantagewhichisattachedtothisproduct BesidestheexchangeofbondsforsharesdiminishestheliabilitiesofthefirmissuerandincreasesinthesametimeitsequitysothatitsdebtcapacityisimprovedFortheconvertiblebondholder Theconvertiblebondisadefensivesecurity verysensitivetoariseinthesharepriceandprotectivewhenthesharepricedecreases Ifthesharepriceincreases theconvertiblepricewillalsoincrease Whensharepricedecreases priceofconvertiblenevergetsbelowthebondfloor i e thepriceofanotherwiseidenticalbulletbondwithnoconversionoption 30 2020 1 27 31 ConvertibleBondsDeterminantsofConvertibleBondPrices Convertiblebondissimilartoanormalcouponbondplusacalloptionontheunderlyingstock Withanimportantdifference theeffectivestrikepriceofthecalloptionwillvarywiththepriceofthebondConvertiblesecuritiesarepricedasafunctionof Thepriceoftheunderlyingstock Expectedfuturevolatilityofequityreturns Riskfreeinterestrates Callprovisions Supplyanddemandforspecificissues Issue specificcorporate Treasuryyieldspread Expectedvolatilityofinterestratesandspreads Thus thereislargeroomforrelativemis valuations 32 ConvertibleBondsConvertibleBondPriceasaFunctionofStockPrice 33 ConvertibleBondsConvertibleBondPricingModel Apopularmethodforpricingconvertiblebondsisthecomponentmodel Theconvertiblebondisdividedintoastraightbondcomponentandacalloptionontheconversionprice withstrikepriceequaltothevalueofthestraightbondcomponent Thefairvalueofthetwocomponentscanbecalculatedwithstandardformulas suchasthefamousBlack Scholesvaluationformula Thispricingapproach however hasseveraldrawbacks First separatingtheconvertibleintoabondcomponentandanoptioncomponentreliesonrestrictiveassumptions suchastheabsenceofembeddedoptions callabilityandputability forinstance areconvertiblebondfeaturesthatcannotbeconsideredintheaboveseparation Second convertiblebondscontainanoptioncomponentwithastochasticstrikepriceequaltothebondprice 34 ConvertibleBondsConvertibleBondPricingModels TheoreticalresearchonconvertiblebondpricingwasinitiatedbyIngersoll 1977 andBrennanandSchwartz 1977 whobothappliedthecontingentclaimsapproachtothevaluationofconvertiblebondsIntheirvaluationmodels theconvertiblebondpricedependsonthefirmvalueastheunderlyingvariableBrennanandSchwartz 1980 extendtheirmodelbyincludingstochasticinterestrates Thesemodelsrelyheavilyonthetheoryofstochasticprocessesandrequirearelativelyhighlevelofmathematicalsophistication 35 ConvertibleBondsBinomialModel ThepriceofthestockonlycangouptoagivenvalueordowntoagivenvalueBesides thereisabond bankaccount thatwillpayinterestofr 36 ConvertibleBondsBinomialModel Weassumeu up d down ForBlackandScholeswewillneedd 1 uForconsistencywealsoneedu 1 r dExample u 1 25 d 0 80 r 10 37 ConvertibleBondsBinomialModel Basicmodelthatdescribesasimpleworld Asthenumberofstepsincreases itbecomesmorerealisticWewillpriceandhedgeanoption itappliestoanyotherderivativesecurityKey wehavethesamenumberofstatesandsecurities completemarkets Basisforarbitragepricing 38 ConvertibleBondsBinomialModel IntroduceanEuropeancalloption K 110Itmaturesattheendoftheperiod 39 ConvertibleBondsBinomialModel WecanreplicatetheoptionwiththestockandthebondConstructaportfoliothatpaysCuinstateuandCdinstatedThepriceofthatportfoliohastobethesameasthepriceoftheoptionOtherwisetherewillbeanarbitrageopportunity 40 ConvertibleBondsBinomialModel WebuysharesandinvestBinthebankTheycanbepositive buyordeposit ornegative shortsellorborrow Wewantthen Withsolution 41 ConvertibleBondsBinomialModel Inourexample wegetforstock And forbonds Thecostoftheportfoliois 42 ConvertibleBondsBinomialModel ThepriceoftheEuropeancallmustbe9 09 Otherwise thereisanarbitrageopportunity Ifthepriceislowerthan9 09wewouldbuythecallandshortselltheportfolioIfhigher theoppositeWehavecomputedthepriceandthehedgesimultaneously Wecanconstructacallbybuyingthestockandborrowing Shortcall theopposite 43 ConvertibleBondsBinomialModel Rememberthat And Substituting 44 ConvertibleBondsBinomialModel Aftersomealgebra Observethecoefficients Positive Smallerthanone AdduptooneLikeaprobability 45 ConvertibleBondsBinomialModel Rewrite Where Thiswouldbethepricingof Ariskneutralinvestor Withsubjectiveprobabilitiespand 1 p 46 ConvertibleBondsBinomialModel Supposethefollowingeconomy WeintroduceanEuropeancallwithstrikepriceKthatmaturesinthesecondperiod 47 ConvertibleBondsBinomialModel Thepriceoftheoptionwillbe Thereare twopaths thatleadtotheintermediatestate thatexplainsthe 2 48 ConvertibleBondsVolatilityintheBinomialModel 49 ConvertibleBondValuationMethodology Giventhataconvertiblebondisnothingbutanoptionontheunderlyingstock weexpecttobeabletousethebinomialmodeltopriceitAteachnode wetest a whetherconversionisoptimal b whetherthepositionoftheissuercanbeimprovedbycallingthebonds Itisadynamicprocedure max min Q1 Q2 Q3 where Q1 valuegivenbytherollback neitherconvertednorcalledback Q2 callprice Q3 valueofstocksifconversiontakesplace 50 ConvertibleBondExample Example 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 or year 51 ConvertibleBondExample Wehave Actually continuouslycompoundedrate 52 ConvertibleBondExample 53 ConvertibleBondExample AtnodeG thebondholderoptimallychoosetoconvertsincewhatisobtainedunderconversion 1 568 31 ishigherthanthepayoffundertheassumptionofnoconversion 1 040 00 ThesameappliestonodeHOntheotherhand atnodesIandJ thevalueundertheassumptionofconversionislowerthanifthebondisnotconvertedtoequity Therefore bondholdersoptimallychoosenottoconvert andthepayoffissimplythenominalvalueofthebond plustheinterestpayments thatis 1 040 00 54 ConvertibleBondExample Workingourwaybackwardthetree weobtainatnodeDthevalueoftheconvertiblebondasthediscountedexpectedvalue usingrisk neutralprobabilitiesofthepayoffsatnodesGandH AtnodeF thesameprincipleapplies exceptthatitanberegardedasastandardbond Wethereforeusetherateofreturnonanonconvertiblebondissuedbythesamecompany 15 55 ConvertibleBondExample AtnodeE thesituationismoreinterestingbecausetheconvertiblebondwillendupasastockincaseofanupmove conversion andasabondincaseofadownmove noconversion Asanapproximateruleofthumb onemayuseaweightedaverageoftheriskfreeandriskyinterestrateinthecomputation wheretheweightingisperformedaccordingtothe risk neutral probabilityofanupversusadownmove Thenthevalueiscomputedas 56 ConvertibleBondExample NotethatatnodeD callingorconvertingisnotrelevantbecauseitdoesnotchangethebondvaluesincethebondisalreadyessentiallyequityAtnodeB itcanbeshownthattheissuerfindsitoptimaltocallthebondIfthebondisindeedcalledbythei