公司理財(cái)（精要版）（第9版）課件 25

Chapter25 OptionValuationMcGraw-Hill/IrwinCopyright?2010byTheMcGraw-HillCompanies,Inc.Allrightsreserved.1KeyConceptsandSkillsUnderstandandbeabletousePut-CallParityBeabletousetheBlack-ScholesOptionPricingModelUnderstandtherelationshipsbetweenoptionpremiumsandstockprice,exerciseprice,timetoexpiration,standarddeviation,andtherisk-freerateUnderstandhowtheoptionpricingmodelcanbeusedtoevaluatecorporatedecisions25-22ChapterOutlinePut-CallParityTheBlack-ScholesOptionPricingModelMoreaboutBlack-ScholesValuationofEquityandDebtinaLeveragedFirmOptionsandCorporateDecisions:SomeApplications25-33ProtectivePutBuytheunderlyingassetandaputoptiontoprotectagainstadeclineinthevalueoftheunderlyingassetPaytheputpremiumtolimitthedownsideriskSimilartopayinganinsurancepremiumtoprotectagainstpotentiallossTrade-offbetweentheamountofprotectionandthepricethatyoupayfortheoption25-44AnAlternativeStrategyYoucouldbuyacalloptionandinvestthepresentvalueoftheexercisepriceinarisk-freeassetIfthevalueoftheassetincreases,youcanbuyitusingthecalloptionandyourinvestmentIfthevalueoftheassetdecreases,youletyouroptionexpireandyoustillhaveyourinvestmentintherisk-freeasset25-55ComparingtheStrategiesStock+PutIfS<E,exerciseputandreceiveEIfS≥E,letputexpireandhaveSCall+PV(E)PV(E)willbeworthEatexpirationoftheoptionIfS<E,letcallexpireandhaveinvestment,EIfS≥E,exercisecallusingtheinvestmentandhaveSValueatExpirationInitialPositionS<ES≥EStock+PutESCall+PV(E)ES25-66Put-CallParityIfthetwopositionsareworththesameattheend,theymustcostthesameatthebeginningThisleadstotheput-callparityconditionS+P=C+PV(E)Ifthisconditiondoesnothold,thereisanarbitrageopportunityBuythe“l(fā)ow”sideandsellthe“high”sideYoucanalsousethisconditiontofindthevalueofanyofthevariables,giventheotherthree25-77Example:FindingtheCallPriceYouhavelookedinthefinancialpressandfoundthefollowinginformation:Currentstockprice=$50Putprice=$1.15Exerciseprice=$45Risk-freerate=5%Expirationin1yearWhatisthecallprice?50+1.15=C+45/(1.05)C=8.2925-88ContinuousCompoundingContinuouscompoundingisgenerallyusedforoptionvaluationTimevalueofmoneyequationswithcontinuouscompoundingEAR=eq-1PV=FVe-RtFV=PVeRtPut-callparitywithcontinuouscompoundingS+P=C+Ee-Rt25-99Example:ContinuousCompoundingWhatisthepresentvalueof$100tobereceivedinthreemonthsiftherequiredreturnis8%,withcontinuouscompounding?PV=100e-.08(3/12)=98.02Whatisthefuturevalueof$500tobereceivedinninemonthsiftherequiredreturnis4%,withcontinuouscompounding?FV=500e.04(9/12)=515.2325-1010PCPExample:PCPwith

ContinuousCompoundingYouhavefoundthefollowinginformation;Stockprice=$60Exerciseprice=$65Callprice=$3Putprice=$7Expirationisin6monthsWhatistherisk-freerateimpliedbytheseprices?S+P=C+Ee-Rt60+7=3+65e-R(6/12).9846=e-.5RR=-(1/.5)ln(.9846)=.031or3.1%25-1111Black-ScholesOption

PricingModelTheBlack-ScholesmodelwasoriginallydevelopedtopricecalloptionsN(d1)andN(d2)arefoundusingthecumulativestandardnormaldistributiontablestddttRESddNEedSNCRtsss-=÷÷????è?++÷???è?=-=-1221212ln)()(25-1212Example:OPMYouarelookingatacalloptionwith6monthstoexpirationandanexercisepriceof$35.Thecurrentstockpriceis$45,andtherisk-freerateis4%.Thestandarddeviationofunderlyingassetreturnsis20%.Whatisthevalueofthecalloption?LookupN(d1)andN(d2)inTable25.3N(d1)=(.9761+.9772)/2=.9767N(d2)=(.9671+.9686)/2=.9679C=45(.9767)–35e-.04(.5)(.9679)C=$10.7525-1313Example:OPMinaSpreadsheetConsiderthepreviousexampleClickontheexcelicontoseehowthisproblemcanbeworkedinaspreadsheet25-1414PutValuesThevalueofaputcanbefoundbyfindingthevalueofthecallandthenusingput-callparityWhatisthevalueoftheputinthepreviousexample?P=C+Ee-Rt–SP=10.75+35e-.04(.5)–45=.06Notethataputmaybeworthmoreifexercisedthanifsold,whileacallisworthmore“alivethandead,”unlessthereisalargeexpectedcashflowfromtheunderlyingasset25-1515Europeanvs.AmericanOptionsTheBlack-ScholesmodelisstrictlyforEuropeanoptionsItdoesnotcapturetheearlyexercisevaluethatsometimesoccurswithaputIfthestockpricefallslowenough,wewouldbebetteroffexercisingnowratherthanlaterAEuropeanoptionwillnotallowforearlyexercise;therefore,thepricecomputedusingthemodelwillbetoolowrelativetothatofanAmericanoptionthatdoesallowforearlyexercise25-1616Table25.425-1717VaryingStockPriceandDeltaWhathappenstothevalueofacall(put)optionifthestockpricechanges,allelseequal?TakethefirstderivativeoftheOPMwithrespecttothestockpriceandyougetdelta.Forcalls:Delta=N(d1)Forputs:Delta=N(d1)-1Deltaisoftenusedasthehedgeratiotodeterminehowmanyoptionsweneedtohedgeaportfolio25-1818WorktheWebExampleThereareseveralgoodoptionscalculatorsontheInternetClickonthewebsurfertogotoandclickontheBasicCalculatorunderAnalysisServicesPricethecalloptionfromtheearlierexampleS=$45;E=$35;R=4%;t=.5;=.2Youcanalsochooseastockandvalueoptionsonaparticularstock25-1919Figure25.1InsertFigure25.1here25-2020Example:DeltaConsiderthepreviousexample:Whatisthedeltaforthecalloption?Whatdoesittellus?N(d1)=.9767ThechangeinoptionvalueisapproximatelyequaltodeltatimesthechangeinstockpriceWhatisthedeltafortheputoption?N(d1)–1=.9767–1=-.0233Whichoptionismoresensitivetochangesinthestockprice?Why?25-2121VaryingTimetoExpiration

andThetaWhathappenstothevalueofacall(put)aswechangethetimetoexpiration,allelseequal?TakethefirstderivativeoftheOPMwithrespecttotimeandyougetthetaOptionsareoftencalled“wasting”assets,becausethevaluedecreasesasexpirationapproaches,evenifallelseremainsthesameOptionvalue=intrinsicvalue+timepremium25-2222Figure25.2Insertfigure25.2here25-2323Example:TimePremiumsWhatwasthetimepremiumforthecallandtheputinthepreviousexample?CallC=10.75;S=45;E=35Intrinsicvalue=max(0,45–35)=10Timepremium=10.75–10=$0.75PutP=.06;S=45;E=35Intrinsicvalue=max(0,35–45)=0Timepremium=.06–0=$0.0625-2424VaryingStandardDeviation

andVegaWhathappenstothevalueofacall(put)whenwevarythestandarddeviationofreturns,allelseequal?TakethefirstderivativeoftheOPMwithrespecttosigmaandyougetvegaOptionvaluesareverysensitivetochangesinthestandarddeviationofreturnThegreaterthestandarddeviation,themorethecallandtheputareworthYourlossislimitedtothepremiumpaid,whilemorevolatilityincreasesyourpotentialgain25-2525Figure25.3Insertfigure25.3here25-2626VaryingtheRisk-FreeRate

andRhoWhathappenstothevalueofacall(put)aswevarytherisk-freerate,allelseequal?ThevalueofacallincreasesThevalueofaputdecreasesTakethefirstderivativeoftheOPMwithrespecttotherisk-freerateandyougetrhoChangesintherisk-freeratehaveverylittleimpactonoptionsvaluesoveranynormalrangeofinterestrates25-2727Figure25.4Insertfigure25.4here25-2828ImpliedStandardDeviationsAlloftheinputsintotheOPMaredirectlyobservable,exceptfortheexpectedstandarddeviationofreturnsTheOPMcanbeusedtocomputethemarket’sestimateoffuturevolatilitybysolvingforthestandarddeviationThisiscalledtheimpliedstandarddeviationOnlineoptionscalculatorsareusefulforthiscomputationsincethereisnotaclosedformsolution25-2929WorktheWebExampleUsetheoptionscalculatorattofindtheimpliedvolatilityofastockofyourchoiceClickonthewebsurfertogototogettherequiredinformationClickonthewebsurfertogotonuma,entertheinformationandfindtheimpliedvolatility25-3030EquityasaCallOptionEquitycanbeviewedasacalloptiononthefirm’sassetswheneverthefirmcarriesdebtThestrikepriceisthecostofmakingthedebtpaymentsTheunderlyingassetpriceisthemarketvalueofthefirm’sassetsIftheintrinsicvalueispositive,thefirmcanexercisetheoptionbypayingoffthedebtIftheintrinsicvalueisnegative,thefirmcanlettheoptionexpireandturnthefirmovertothebondholdersThisconceptisusefulinvaluingcertaintypesofcorporatedecisions25-3131ValuingEquityandChanges

inAssetsConsiderafirmthathasazero-couponbondthatmaturesin4years.Thefacevalueis$30million,andtherisk-freerateis6%.Thecurrentmarketvalueofthefirm’sassetsis$40million,andthefirm’sequityiscurrentlyworth$18million.SupposethefirmisconsideringaprojectwithanNPV=$500,000.Whatistheimpliedstandarddeviationofreturns?Whatisthedelta?Whatisthechangeinstockholdervalue?25-3232PCPandtheBalance

SheetIdentityRiskydebtcanbeviewedasarisk-freebondminusthecostofaputoptionValueofriskybond=Ee-Rt–PConsidertheput-callparityequationandrearrangeS=C+Ee-Rt–PValueofassets=valueofequity+valueofariskybondThisisjustthesameasthetraditionalbalancesheetidentityAssets=liabilities+equity25-3333MergersandDiversificationDiversificationisafrequentlymentionedreasonformergersDiversificationreducesriskand,therefore,volatilityDecreasingvolatilitydecreasesthevalueofanoptionAssumediversificationistheonlybenefittoamergerSinceequitycanbeviewedasacalloption,shouldthemergerincreaseordecreasethevalueoftheequity?Sinceriskydebtcanbeviewedasrisk-freedebtminusaputoption,whathappenstothevalueoftheriskydebt?Overall,whathashappenedwiththemergerandisitagooddecisioninviewofthegoalofstockholderwealthmaximization?25-3434ExtendedExample–PartIConsiderthefollowingtwomergercandidatesThemergerisfordiversificationpurposesonlywithnosynergiesinvolvedRisk-freerateis4%CompanyACompanyBMarketvalueofassets$40million$15millionFacevalueofzerocoupondebt$18million$7millionDebtmaturity4years4yearsAssetreturnstandarddeviation40%50%25-3535ExtendedExample–PartIIUsetheOPM(oranoptionscalculator)tocomputethevalueoftheequityValueofthedebt=valueofassets–valueofequityCompanyACompanyBMarketValueofEquity25.6819.867MarketValueofDebt14.3195.13325-3636ExtendedExample–PartIIITheassetreturnstandarddeviationforthecombinedfirmis30%Marketvalueassets(combined)=40+15=55Facevaluedebt(combined)=18+7=25CombinedFirmMarketvalueofequity34.120Marketvalueofdebt20.880TotalMVofequityofseparatefirms=25.681+9.867=35.548Wealthtransferfromstockholderstobondholders=35.548–34.120=1.428(exactincreaseinMVofdebt)25-3737M&AConclusionsMergersfordiversificationonlytransferwealthfromthestockholderstothebondholdersThestandarddeviationofreturnsontheassetsisreduced,therebyreducingtheoptionvalueoftheequityIfmanagement’sgoalistomaximizestockholderwealth,thenmergersforreasonsofdiversificationshouldnotoccur25-3838ExtendedExample:

LowNPV–PartIStockholdersmaypreferlowNPVprojectstohighNPVprojectsifthefirmishighlyleveragedandthelowNPVprojectincreasesvolatilityConsideracompanywiththefollowingcharacteristicsMVassets=40millionFaceValuedebt=25millionDebtmaturity=5yearsAssetreturnstandarddeviation=40%Risk-freerate=4%25-3939ExtendedExample:

LowNPV–PartIICurrentmarketvalueofequity=$22.657millionCurrentmarketvalueofdebt=$17.343millionProjectIProjectIINPV$3$1MVofassets$43$41Assetreturnstandarddeviation30%50%MVofequity$23.769$25.339MVofdebt$19.231$15.66125-4040ExtendedExample:

LowNPV–PartIIIWhichprojectshouldmanagementtake?EventhoughprojectBhasalowerNPV,itisbetterforstockholdersThefirmhasarelativelyhighamountofleverageWithprojectA,thebondholdersshareintheNPVbecauseitreducestheriskofbankruptcyWithprojectB,thestockholdersactuallyappropriateadditionalwealthfromthebondholdersforalargergaininvalue25-4141ExtendedExample:

NegativeNPV–PartIWe’veseenthatstockholdersmightpreferalowNPVtoahighone,butwouldtheyeverpreferanegativeNPV?Undercertaincircumstances,theymightIfthefirmishighlyleveraged,stockholdershavenothingtoloseifaprojectfailsandeverythingtogainifitsucceedsConsequently,theymaypreferaveryriskyprojectwithanegativeNPVbuthighpotentialrewards25-4242ExtendedExample:

NegativeNPV–PartIIConsiderthepreviousfirmTheyhaveoneadditionalprojecttheyareconsideringwiththefollowingcharacteristicsProjectNPV=-$2millionMVofassets=$38millionAssetreturnstandarddeviation=65%EstimatethevalueofthedebtandequityMVequity=$25.423millionMVdebt=$12.577million25-4343ExtendedExample:

NegativeNPV–PartIIIInthiscase,stockholderswouldactuallypreferthenegativeNPVprojecttoeitherofthepositiveNPVprojectsThestockholdersbenefitfromtheincreasedvolatilityassociatedwiththeprojecteveniftheexpectedNPVisnegativeThishappensbec