《金融学》（第二版） 课件 （英文） -Chapter15-17 Markets for Options and Contingent Claims；Financial Structure of the Firm；Real Options

Chapter15:

MarketsforOptions andContingentClaimsObjectiveOptionsPricingrelationshipsPricingmodelsFinancialdecisionsanalyzedthroughoptions1Chapter15Contents15.1HowOptionsWork15.2InvestingwithOptions15.3ThePut-CallParityRelationship15.4Volatility&OptionPrices15.5Two-State(Binomial)OptionPricing15.6DynamicReplication&theBinomialModel15.7TheBlack-ScholesModel15.8ImpliedVolatility15.9ContingentClaimsAnalysisofCorporateDebtandEquity15.10CreditGuarantees15.11OtherApplicationsofOption-PricingMethodology2ObjectivesHowtouseoptionstomodifyone’sexposuretoinvestmentrisk.Tounderstandthepricingrelationshipsthatexistamongcalls,puts,stocks,andbonds.ToexplainthebinomialandBlack-Scholesoption-pricingmodelsandapplythemtothevaluationofcorporatebondsandothercontingentclaims.Toexploretherangeoffinancialdecisionsthatcanbefruitfullyanalyzedintermsofoptions.3IntroductionThischapterexploreshowoptionpricesareaffectedbythevolatilityoftheunderlyingsecurityExchangetradedoptionsappearedin1973,enablingustodeterminethemarket’sestimateoffuturevolatility,ratherthanrelyingonhistoricalvalues4DefinitionofanOptionRecallthatanAmerican{European}call(put)optionistheright,butnottheobligationtobuy(sell)anassetataspecifiedpriceanytimebeforeitsexpirationdate{onitsexpirationdate}5UbiquitousOptionsThischapterfocusesontradedoptions,butitwouldbeamistaketobelievethatthetoolswewillbedevelopingarerestrictedtotradedoptionsSomeexamplesofoptionsaregivenonthenextfewslides6GovernmentPriceSupportsGovernmentssometimesprovideassistancetofarmersbyofferingtopurchaseagriculturalproductsataspecifiedsupportpriceIfthemarketpriceislowerthanthesupport,thenafarmerwillexerciseherrightto‘put’hercroptothegovernmentatthehigherprice7OldMortgageTraditionalUSmortgagesgivethehouseholdertherighttocallthemortgageatastrikeequaltotheoutstandingprincipleIfinterestrateshavefallenbelowthenote’srate,thenthehomeownerwillconsiderrefinancingthemortgage8NewMortgageYoupaysome‘points’tolock-inaninterestrateonamortgageIfratesfall,youmayrenegotiatethemortgage,andthenpaymorepointstolockinthenewlowerrateIfratesrise,thenyouwillgotothesettlementtablewithalower-than-marketinterestrate9TenureandSeniorityInacompanythathasapolicyoflast-infirst-out,aworkerwithsenioritymayforegoahighersalaryinanothercompanybecauseofthelossofjobsecurityTheworkerhasbeengiventheright,butnottheobligation,tohaveworkunderasetofadverseeconomicconditions10CopperPennies,SilverCoinSilverandcoppercoinagehasbeenreplacedbyzincandcooperalloys/compositestoreducetheirmintingcostsTheoldcoinsareoftenlegalcurrency,andsocontainanoptionfeature:Ifthepriceoftheunderlyingmetalfallsbelowitslegalvalue,Ihavetherighttoreturnthecoinintocirculation11InsuranceInsurancepoliciesoftengiveyoutheright,butnottheobligationtodosomething,itisthereforeoption-likeTherenewablerideronatermlifepolicyisanoptionIfsomebody:isterminallyill,thentheriderisveryvaluableremainsingoodhealth,thenitisnotvaluable12SupplyContractsAnuclearpowerplantsupplieroncegotintoserioustroublebyguaranteeingtosupplyenricheduraniumatafixedpriceThemarketpriceofenricheduraniumroseprecipitously13TechnologicalLeasesAcomputerleasingcompanyhadaclauseinitsleasestatingthatthecustomerhadtherighttocancelThecomputermanufacturerintroducednextgenerationofcomputers,andtheleasingcompany’scustomer’scanceledtheirleases,resultinginamassiveinventoryofobsoletecomputers14LimitedLiabilityTheownersofalimitedliabilitycorporationhavetheright,butnottheobligation,to‘put’thecompanytothecorporation’screditorsandbondholdersLimitedliabilityis,ineffect,aputoption15TradingonCommissionYouareatraderwithacontractgivingyouacommissionof20%ofeachmonthstradingprofitsIfyoumakealoss,thenyouwalkaway,butifyoumakeaprofit,youstayYoumaybetemptedtoincreaseyourvolatilitytoboostthevalueofyouroption1615.1HowOptionsWorkTheLanguageofOptionsContingentClaim:Anyassetwhosefuturepay-offdependsupontheoutcomeofanuncertaineventCall:anoptiontobuyPut:anoptiontosellStrikeorExercisePrice:thefixedpricespecifiedinanoptioncontractExpirationorMaturityDate:Thedateafterwhichanoptioncan’tbeexercisedAmericanOption:anoptionthatcanbeexercisedatanytimeuptoandincludingmaturitydate17EuropeanOption:anoptionthatcanonlybeexercisedonthematuritydateTangibleValue:ThehypotheticalvalueofanoptionifitwereexercisedimmediatelyAt-the-Money:anoptionwithastrikepriceequaltothevalueoftheunderlyingassetOut-of-the-Money:anoptionthat’snotat-the-money,buthasnotangiblevalueIn-the-Money:anoptionwithatangiblevalueTimeValue:thedifferencebetweenanoption’smarketvalueanditstangiblevalueExchange-TradedOption:AstandardizedoptionthatanexchangestandsbehindinthecaseofadefaultOvertheCounterOption:Anoptiononasecuritythatisnotanexchange-tradedoption18192015.2InvestingwithOptionsThepayoffdiagram(terminalconditions,boundaryconditions)foracallandaputoption,eachwithastrike(exerciseprice)of$100,isderivednext21OptionPayoffDiagramsThevalueofanoptionatexpirationfollowsimmediatelyfromitsdefinitionInthecaseofacalloptionwithstrikeof$100,ifthestockpriceis$90($110),thenexercisingtheoptionresultspurchasingthesharefor$100,whichis$10moreexpensive($10lessexpensive)thanbuyingit,soyouwouldn't(would)exerciseyourright22

CallOptionPayoffDiagram23PutOptionPayoffDiagram24PayoffDiagramsforAlternativeBullishStockStrategies252615.3ThePut-CallParityRelationConsiderthefollowingtwostrategiesPurchaseaputwithastrikepriceof$100,andtheunderlyingsharePurchaseacallwithastrikepriceof$100andabondthatmaturesatthesamedatewithafaceof$100Thematurityvaluesaretabulatedandplottedagainstshareprice:27PayoffDiagramforPureDiscountBondPlusCall282930ObservationThemostimportantpointtoobserveisthatthevalueofthe“call+bond”strategy,isidentical(atmaturity)withtheprotective-putstrategy“put+share”So,iftheputandthecallhavethesamestrikeprice,weobtaintheput-callparityrelationship:put+share=call+bond31TechnicalNoteTheaboverelationshipistrue,ingeneral,fordividend-lessEuropeanoptions,buttheactualproofrequirestakingexpectationsoftheoptionandsecurityboundaryconditionsTheargumentisthereforeaheuristicforrememberingand‘seeing’therelationshipThefullproofisleftforyourinvestmentclass32Put-CallParityforAmericanandEuropeanOptionsAEuropeanoptionthatpaysnodividendduringitslifefullysatisfiestherequirementsofput-callparityInthecaseofAmericanoptions,therelationshipisfullyaccurateonlyatmaturity,becauseAmericanputsaresometimesexercisedearly33Put-CallParityEquation34SyntheticSecuritiesTheput-callparityrelationshipmaybesolvedforanyofthefoursecurityvariablestocreatesyntheticsecurities:C=S+P-BS=C-P+BP=C-S+BB=S+P-C35SyntheticSecuritiesC=S+P-BandP=C-S+BmaybeusedbyfloortraderstoflipbetweenacallandaputS=C-P+Bmaybeusedbyshort-termtraderswishingtotakeadvantageoflowertransactioncostsB=S+P-Cmaybeusedtocreateasyntheticbondsaidtopayaslightlyhigherreturnthanthephysicalbond36OptionsandForwardsWesawinthelastchapterthatthediscountedvalueoftheforwardwasequaltothecurrentspotTherelationshipbecomes37ImplicationsforEuropeanOptionsIftheforwardpriceoftheunderlyingstockisequaltothestrikeprice,thenthevalueofthecallisequaltothevalueoftheputThisrelationshipissoimportant,thatsomeoptiontradersdefine‘at-the-money’intermsoftheforwardratherthanthespot38ImplicationsforEuropeanOptionsIf(F>E)then(C>P)If(F=E)then(C=P)If(F<E)then(C<P)EisthecommonstrikepriceFistheforwardpriceofunderlyingshareCisthecallpricePistheputprice39Strike=ForwardCall=Put4041PVStrikeStrike4215.4VolatilityandOptionPrices

WenextexplorewhathappenstothevalueofanoptionwhenthevolatilityoftheunderlyingstockincreasesWeassumeaworldinwhichthestockpricemovesduringtheyearfrom$100tooneoftwonewvaluesattheendoftheyearwhentheoptionmaturesAssumeriskneutrality4344IllustrationExplainedThestockvolatilityinthesecondscenarioishigher,andtheexpectedpayoffsforboththeputandthecallarealsohigherThisistheresultoftruncation,andholdsinallempiricallyreasonablecasesConclusion:Volatilityincreasesalloptionprices4515.5Two-State(Binomial)Option-PricingWearenowgoingtoderivearelativelysimplemodelforevaluatingoptionsTheassumptionswillatfirstappeartotallyunrealistic,butusingsomeunderhandmathematics,themodelmaybemadetopriceoptionstoanydesiredlevelofaccuracyTheadvantageofthemethodisthatitdoesnotrequirelearningstochasticcalculus,andyetitillustratesallthekeystepsnecessarytoderiveanyoptionevaluationmodel46BinaryModelAssumptionsAssume:theexercisepriceisequaltotheforwardpriceoftheunderlyingstockoptionpricesthendependonlyonthevolatilityandtimetomaturity,anddonotdependoninterestratestheputandcallhavethesameprice47BinaryModelAssumptionsMorespecificallyweassume:shareprice=strikeprice=$100timetomaturity=1yeardividendrate=interestrate=0stockpriceseitherriseorfallby20%intheyear,andsoareeither$80or$120atyearend48BinaryModel:CallStrategy:replicatethecallusingaportfoliooftheunderlyingstocktherisklessbondbythelawofoneprice,thepriceoftheactualcallmustequalthepriceofthesyntheticcall49BinaryModel:CallImplementation:thesyntheticcall,C,iscreatedbybuyingafractionxofshares,ofthestock,S,andsimultaneouslysellingshortriskfreebondswithamarketvalueythefractionxiscalledthehedgeratio50BinaryModel:CallSpecification:Wehaveanequation,andgiventhevalueoftheterminalshareprice,weknowtheterminaloptionvaluefortwocases:Byinspection,thesolutionisx=1/2,y=4051BinaryModel:CallSolution:Wenowsubstitutethevalueoftheparametersx=1/2,y=40intotheequationtoobtain:52BinaryModel:PutStrategy:replicatetheputusingaportfoliooftheunderlyingstockandrisklessbondbythelawofoneprice,thepriceoftheactualputmustequalthepriceofthesyntheticputreplicatedaboveMinorchangestothecallargumentaremadeinthenextfewslidesfortheput53BinaryModel:PutImplementation:thesyntheticput,P,iscreatedbysellshortafractionxofshares,ofthestock,S,andsimultaneouslybuyriskfreebondswithamarketvalueythefractionxiscalledthehedgeratio54DecisionTreeforDynamicReplication

ofCallOption55BinaryModel:PutSpecification:Wehaveanequation,andgiventhevalueoftheterminalshareprice,weknowtheterminaloptionvaluefortwocases:Byinspection,thesolutionisx=1/2,y=6056BinaryModel:PutSolution:Wenowsubstitutethevalueoftheparametersx=1/2,y=60intotheequationtoobtain:5715.6DynamicReplicationandtheBinomialModelWenowtakethenextsteptowardsgreaterrealismbydividingtheyearinto2sub-periodsofhalfayeareach.Thisgives3possibleoutcomesOurfirsttaskistofindaself-financinginvestmentstrategythatdoesnotrequireinjectionorwithdrawalofnewfundsduringthelifeoftheoptionWefirstcreateadecisiontree:58DecisionTreeforDynamicReplicationofaCallOption($120*100%)+(-$100)=$2059ReadingtheDecisionTreeThetreeisconstructedbackwardsbecauseweknowonlythefuturecontingentcallpricesForExample,whenconstructingtheweightsfortime6-months,theoptionpricesfor12-monthsareusedForconsistencywiththenextmodel,thediscretestockpricesareusuallyfixedratios,i.e.121,110,100,90.91,82.6460ThePowerofLatticeModelsLatticemodels,ofwhichthebinarymodelisthesimplest,areveryimportanttotradersbecausetheymaybemodifiedtohandledifferentdistributions,thepossibilityofearlyexercise,anddiscretedividendpaymentsToseehoweasyitistochangethedistributionalassumption,theaboveillustrationresultsinstockpricesbeingnormallydistributed,andthemodificationresultsinalognormaldistributionofprices6115.7TheBlack-ScholesModel

ThemostwidelyusedmodelforpricingoptionsistheBlack-ScholesmodelThismodeliscompletelyconsistentwiththebinarymodelastheintervalbetweenstockpricesdecreasestozeroThemodelprovidestheoreticalinsightsintooptionbehaviorTheassumptionsareelegant,simple,andquiterealistic62TheBlack-ScholesModel

WewillworkwiththegeneralizedformofthemodelbecausethesmalladditionalcomplexityresultsinconsiderableadditionalpowerandflexibilityFirst,notation:63TheBlack-ScholesModel:Notation

C=priceofcallP=priceofputS=priceofstockE=exercisepriceT=timetomaturityln(.)=naturallogarithme=2.71828...N(.)=cum.norm.dist’nThefollowingareannual,compoundedcontinuously:r=domesticriskfreerateofinterestd=foreignriskfreerateorconstantdividendyieldσ=volatility64TheNormalProblemItisnotunusualforastudenttohaveaproblemcomputingthecumulativenormaldistributionusingtablestablestructuresvary,sobecarefulusingstandard-issuenormaltablesdegradescomputedoptionvaluesbecauseoferrorscausedbycatastrophicsubtraction{ManyprofessionalsuseHasting’sformulaasreportedinAbramowitzandStegunasequation26.2.19(never,neveruse26.2.18).Itscertificatevalidin0<=x<Inf,sousesymmetrytoget-Inf<x<0}65TheNormalProblemThefunctionsthatcomewithExcelhaveadequateaccuracy,soconsiderusing‘Normsdist()’inthestatisticalfunctions(notethesinNormsdist)66TheBlack-ScholesModel:What’smissingTherearenoexpectationsaboutfuturereturnsinthemodelThemodelispreference-freeσ-risk,notb-risk,istherelevantrisk67TheBlack-ScholesModel:Equations68TheBlack-ScholesModel:Equations(ForwardForm)69TheBlack-ScholesModel:Equations(Simplified)70SoWhatDoesitMean?Youcannowobtainthevalueofnon-dividendpayingEuropeanoptionsWithalittleskill,youcanwidenthistoobtainapproximatevaluesofEuropeanoptionsonsharespayingadividend,andtosomeAmericanoptions717273ImpliedVolatilitySPX

(July2005-June2006)7475ObservableVariablesAllthevariablesaredirectlyobservable,exceptingthevolatility,σ,andpossibly,thenextcashdividend,dWedonothavetodelveintothepsycheofinvestorstoevaluateoptionsWedonotforecastfuturepricestoobtainoptionvalues7615.8ImpliedVolatilityThefollowingslidesshowhowtoestimatevolatilityusingExcelTheoptionmostcommonlyusedtoestimatevolatilityistheoneclosesttothepresentvalueofthestrikeprice:Thatis,theoptionthatishasastrikeclosesttotheforwardpriceThisoptionhasthemost“oomph”77InsertanynumbertostartFormulaforoptionvalueminustheactualcallvalue7879Pat’sPlanPathasaplantogetrichwithnorisk:Setupspecialportfolio,(Patcallsthisa“selffinancing,deltaneutralportfoliowithpositivecurvature,”butPathasthisthingwithwords)80Pat,theStrategistShortsomeshares,andoff-setsmallpricechangesaboutthecurrentpricewithsomecalloptions,theninvestthedifferenceinbonds8182Pat,theCartographerApparently,whatPathasdoneistofindthetangent(attoday’sshareprice)ofthecallvaluecurve,usingbondsandstockintherightproportionsThisiswhatwedidearlierwhenweconstructedthebinarypricingmodelAtthecurrentpriceof$103,thetangentmimicsthecallcurve83Pat(Continued)Patthenwentshortthetangencyportfolio,andlongthecalltocreatethethickblackportfolioObservethattheminimumvalueoftheportfolioiszero,andthisoccursatthecurrentprice,soitisself-financingPatmakesmoneyifsharepricesmoveupordown84PatTriumphantThisclearlydefeatsthelawofoneprice:Thereisnodownsiderisk,noconstructioncosts,andyetwillyieldapositiveprofitalmostallthetimeWhat’swrongwithPat’sanalysis?85PatDejectedTheansweristhatittakestimeforapricetomove,andduringthattime,allotherthingsbeingequal,thevalueoftheoptionwilldecayThinkofadownwardssloping,veryslick,rain-guttercontainingacritter:Thecrittermayclimbthewallsofthegutter,butitisconstantlyslidingdownthegutter86PatinDespairThenextdiagramshowsthevalueoftheportfoliotodayandoneweekhenceTheconstructionlineshavebeenremoved,andthegraphhasbeenre-scaled8788PatCondemnedtoPovertyThediagramshowsthat,ifnextweek’ssharepricesfallbetweenabout$97and$105.5,PatwillenjoyalossAstimepasses,decaywillmakethisstrategyaveryriskyoneAnotherfactorPatdidnottakeintoaccountisthatvolatilityisitselfvolatile,sothehedgemaydisintegrate8915.9ContingentClaimsAnalysisofCorporateDebtandEquityTheCCAapproachusesadifferentsetofinformationalassumptionsthanthediscountedcashflow(DCF)method:itusestherisk-freerateratherthanarisk-adjusteddiscountrateitusesknowledgeofthepricesofoneormorerelatedassetsandtheirvolatilities90Contingent-ClaimsAnalysisofStockandBonds:DebtcoDebtcoisareal-estateholdingcompanyandhasissued1,000,000commonshares80,000purediscountbonds,face$,1000,maturity1-year91Debtco,ContinuedThetotalmarketvalueofDebtcois$100,000,000Therisk-freerate,(andtherefore,bythelawofoneprice,Debtco’sbondrate,)is4%92Debtco,NotationEthemarketvalueofthestockissueDthemarketvalueofthedebtissueVthetotalcurrentmarketvalue;V=E+DV1thetotalmarketvalueoneyearhence(ThelawofonepriceensuresthatV=E+Dmustbetrue,otherwisetherewillbeanarbitrageopportunity)93Debtco,SecurityValuationValueofthebondsBytheruleofoneprice,thevalueofthebondsmustequaltheirfacevaluediscountedattherisk-freerateforayearD=80,000*$1,000/1.04=$76,923,077Bythetotalvalueofthefirm,V=E+D,thevalueofthestockisE=V-D=$100,000,000-$76,923,077=$23,076,92394Debtco,PayoffAconsequenceofDebtco’shavingbondswitharisk-freerateisthatthecompanyhaseitherpurchasedbonddefaultinsurancefromathirdparty,orthatthefirm’sassetshaveno(downside)riskFormanycompanies,amorerealisticassumptionisthattheassetsdohaverisk,andtoevaluatesuchsecuritiesrequiresapayofffunctionforthebondsorstock:9596NegativeFirmValuesWehaveassumedthatthevalueofthefirmneverfallsbelowzero,butwhileunusual,itispossibleforthemarketvalueofafirm’sassetstobelessthanzeroConsiderEnviromessInc.,afirmthatforyearspollutedtheHudsonRiverwithabyproductofLifecide®Thecostofcleaninguptherivermaywellgreatlyexceedthefirm’sfinancialresources97NegativeFirmValuesNegativevaluesoflimitedliabilitycompaniesisirrelevanttotheconstructionofthepayoffdiagramsItdoesinfluencethevalueofthefirm’sdebtandequitythroughthe(truncated)distributionofthefirmsfuturevalues98Debtco,ProbabilitiesInadditiontothepayoffdiagrams,weneedinformationabouttheprobabilitiesofthefuturevaluesofthefirm99ProbabilityDensityofaFirm'sValue0.000.010.020.030.040.050.060.070.080.09020406080100120140160180200ValueofaFirmProbabilityDensity100Debtco,ProbabilitiesConsiderableeffortisemployedwhenestimatingtheprobabilitiesusedinCCA,but,toobtainabasicunderstandingofCCAitisenoughtoassumeaverysimpledistributionWewillassumethatthefirmmaytakeononlyoneoftwopossiblevaluesayearfromnow,whenthebondmatures,namely$70or$140million(Thetwo-stateassumptioncanbegeneralizedintoan-statelatticemodelwithanyspecifieddegreeofaccuracy)101DebtcoSecurityPayoffTable($’000,000)102Debtco’sReplicatingPortfolioLetxbethefractionofthefirminreplicatorYbetheborrowingsattherisk-freerateinthereplicatorIn$’000,000thefollowingequationsmustbesatisfied103Debtco’sReplicatingPortfolio($’000)104Debtco’sReplicatingPortfolioNotethattheaboveslideshowsthat(withtheweightsonthelast-but-oneslide)thevalueofthereplicatingportfolioandthestockareidenticalnowandone-yearhenceBythelawofoneprice,thevalueofDebtco’sStockis$28.02each(Onemillionissued)105Debtco’sReplicatingPortfolioWeknowvalueofthefirmis$1,000,000,andthevalueofthetotalequityis$28,021,978,sothemarketvalueofthedebtwithafaceof80,000,000is$71,978,022Theyieldonthisdebtis(80…/71…)-1=11.14%106AnotherViewofDebtco’sReplicatingPortfolio(‘$000)107InterpretationThemarketvalueofthefirm’sriskydebtconsistsofabout$58millionrisklessdebtandabout$14millionoftheriskyfirmThereisasensethatholdersofriskydebtacceptsomeofthefirm-as-a-whole’sriskycashflow,justasshareholdersdoShareholdersaccepttheremaining$85millionofthefirm-as-a-whole,andfinanceitwiththeequivalentofabout$58millionindefault-freedebt108ValuingtheBondsgiventhePriceoftheStockTherearethreevaluesthatareinequilibriumwitheachother:thevalueofthefirm(Ödone)thevalueofthebond(tobedonenext)thevalueofthestockIfweknowone,theothersmaybededuced109ValuingBondsGiventheStockPrice($’000,000’s)Assumethat,asbeforewehave:Scenarioa:valueoffirmin1-year=$70Scenariob:valueoffirmin1-year=$140Risk-free1-yearbondsproducea4%yieldTotalfacevalueofDebtco’sbondsis$80Assumealso:Debtcohasamillionsharesoutstanding,totalmarketvalue$20110ValuingBondsWecanreplicatethefirm’sequityusingx=6/7ofthefirm,andaboutY=$58millionrisklessborrowing(earlieranalysis)Theimpliedvalueofthebondsisthen$90,641,026-$20,000,000=$70,641,026&theyieldis(80.00-70.64)/70.64=13.25%111ValuingStockGiventheBondYieldAssumethat,asbeforewehave:Scenarioa:valueoffirmin1-year=$70Scenariob:valueoffirmin1-year=$140Risk-free1-yearbondsproducea4%yieldAssumealso:Debtco’sbondsyield10%(currentvalue$909.09)112BondReplicationInordertoreplicatethebond,wewillpurchaseafraction,x,ofthefirm,andpurchasethevalueYofrisk-freebondsAtmaturity,thevalueofthebondsisScenarioa,V=$70million:$70millionScenariob,V=$140million:$80million113ReplicationPortfolio114DeterminingtheWeightofFirmInvestedinBond,x,andtheValueoftheR.F.-Bond,Y115ValuingStockWecanreplicatethebondbypurchasing1/7ofthecompany,and$57,692,308ofdefault-free1-yearbondsThemarketvalueofthebondsis$909.0909*80,000=$72,727,273ThevalueofthestockisthereforeE=V-D=$105,244,753-$72,727,273=$32,517,480116ConvertibleBonds

Aconvertiblebondobligatestheissuingfirmtoredeemthebondatparvalueuponmaturity,ortoallowthebondholdertoconvertthebondintoapre-specifiednumberofshareofcommonstock117ConvertibleBonds:TheConvertidebtCorporation

AssumethatConvertidebtisineverywaylikeDebtco,buteachbondisconvertibleto20commonstockatmaturityIfallthedebtisconverted,thenthenumberofcommonstockwillrisefrom1,000,000to1,000,000+80,000*20=2,600,000shares118119BondholderEntitlementsGiventhataconversionoccurs,thevalueofeachcommonstockwillbeValueoffirm/2,600,000Thebondholderswillreceive1,600,000oftheseshares,sothebondholderswillown1.6/2.6ofthefirm,leavingtheshareholderswith1/2.6ofthefirmThecriticalvalueforconversionisfirm’svalue=80million*2.6/1.6=$130million120PayoffsfromConvertibleBondsScenarioa:Thevalueofthefirmis70million:Thebondholderswillownthecompany,sothevalueis70millionScenariob:Thevalueofthefirmis140million:Thebondholderswillown1.6/2.6ofthecompany,or$86,153,846Theremaininganalysisfollowsexactlythesamerecipeastheconventionalbond121DynamicReplication

Ifyoureferbacktotheconvertiblebondexample,youwillobservethatonlytwopointsonthedoubly-kinkedpayoffcurveweresampled(clearlyunrealistic)Asinthecaseofbinaryoptionevaluation,byincreasingthenumberofsamplepoints,youmayachieveanydesiredlevelofaccuracy122OutlineofMethodBifurcatetheyearintotwosix-monthintervalsStartingatthepresenttimeatNode-A(valueofthefirmis$100million)therearetwoscenariosthepricerisesto$115MM(Node-B)thepricefallsto$90MM(Node-C)123SetupAssumptions(SeeNextDiagram)GivenNode-Boccurredatmonth6($115MM),thenafterafurther6-months,therearetwomorescenariosthepricerisesto$140MM(Node-D)thepricefallsallthewayto$90MM(Node-E)GivenNode-Coccurredatmonth6($90MM),thenafterafurther6-months,therearetwomorescenariosThepricerisesto$110MM(Node-F)Thepricefallsfurtherto$70MM(Node-G)124OutlineDecisionTreeNode-B$115MMNode-C$90MMNode-D$140MMNode-F$110MMNode-E$90MMNode-G$70MMNode-A$100MMMonth0Month6Month12125OutlineofMethodTherearethreedecisionnodesA,B,CAteachnode,areplicatingdecisionismadeFirstB(fromD&E),andindependentlyC(fromF&G)Then,usingthebackwardsinduction,A(fromB&C)Thestepsareexactlyasoutlinedindetailforanon-compositedecisionsTheexampleisfeaturedinthetextbook126OutlineofMethodTheonlydetailthatrequiresyourattentionisthatth