《金融学第二版》讲义大纲及课后习题答案详解 第15章.docx

《金融学第二版》讲义大纲及课后习题答案详解 第15章.docx

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《金融学第二版》讲义大纲及课后习题答案详解第15章

CHAPTER15

OPTIONSANDCONTINGENTCLAIMS

Objectives

∙Howtouseoptionstomodifyone’sexposuretoinvestmentrisk.

∙Tounderstandthepricingrelationshipsthatexistamongcalls,puts,stocksandbonds.

∙ToexplainthebinomialandBlack-Scholesoption-pricingmodelsandapplythemtothevaluationofcorporatebondsandothercontingentclaims.

∙Toexploretherangeoffinancialdecisionsthatcanbefruitfullyanalyzedintermsofoptions.

Outline

15.1HowOptionsWork

15.2InvestingwithOptions

15.3ThePut-CallParityRelation

15.4VolatilityandOptionPrices

15.5Two-State（Binomial）Option-Pricing

15.6DynamicReplicationandtheBinomialModel

15.7TheBlack-ScholesModel

15.8ImpliedVolatility

15.9ContingentClaimsAnalysisofCorporateDebtandEquity

15.10CreditGuarantees

15.11OtherApplicationsofOption-PricingMethodology

Summary

∙Optionscanbeusedtomodifyaninvestor’sexposuretoinvestmentrisk.Bycombiningtherisk-freeassetandstock-indexcalloptions,aninvestorcanachieveaguaranteedminimumrateofreturnplussubstantialupsideparticipationinthestockmarket.

∙AportfolioconsistingofastockplusaEuropeanputoptionisequivalenttoarisk-freebondwithafacevalueequaltotheoption’sexercisepriceplusaEuropeancalloption.ThereforebytheLawofOnePrice,wegettheput-callparityrelation:

whereSisthestockprice,Pthepriceoftheput,rtherisk-freeinterestrate,Tthematurityoftheoption,andCthepriceofthecall.

∙Onecancreateasyntheticoptionfromtheunderlyingstockandtherisk-freeassetthroughadynamicreplicationstrategythatisself-financingaftertheinitialinvestment.BytheLawofOnePrice,theoption’spriceisgivenbytheformula:

where:

C=priceofthecall

S=priceofthestock

E=exerciseprice

r=risk-freeinterestrate（theannualizedcontinuouslycompoundedrateonasafeassetwiththesamematurityastheoption）

T=timetomaturityoftheoptioninyears

=standarddeviationoftheannualizedcontinuouslycompoundedrateofreturnonthestock

d=continuousdividendyieldonthestock

ln=naturallogarithm

e=thebaseofthenaturallogfunction（approximately2.71828）

N（d1）=theprobabilitythatarandomdrawfromastandardnormaldistributionwillbelessthand1.

∙Thesamemethodologyusedtopriceoptionscanbeusedtovaluemanyothercontingentclaims,includingcorporatestocksandbonds,loanguarantees,andthe“realoptions”imbeddedininvestmentsinresearchanddevelopmentandflexiblemanufacturingtechnology.

SolutionstoProblemsatEndofChapter

PayoffDiagrams

1.GraphthepayoffforaEuropeanputoptionwithexercisepriceE,writtenonastockwithvalueS,when:

a.Youholdalongposition（i.e.,youbuytheput）

b.Youholdashortposition（i.e.,youselltheput）

SOLUTION:

a.

b.

2.GraphthepayofftoaportfolioholdingoneEuropeancalloptionandoneEuropeanputoption,eachwiththesameexpirationdateandeachwithexercisepriceE,whenbothoptionsareonastockwithvalueS.

SOLUTION:

Investingwithoptions

3.Therisk-freeone–yearrateofinterestis4%,andtheGlobalexstockindexisat100.Thepriceofone-yearEuropeancalloptionsontheGlobalexstockindexwithanexercisepriceof104is8%ofthecurrentpriceoftheindex.AssumethattheexpecteddividendyieldonthestocksintheGlobalexindexiszero.Youhave$1milliontoinvestforthenextyear.Youplantoinvestenoughofyourmoneyinone-yearT-billstoinsurethatyouwillatleastgetbackyouroriginal$1million,andyouwillusetherestofyourmoneytobuyGlobalexcalloptions.

a.AssumingthatyoucaninvestfractionalamountsinGlobalexoptions,showthepayoffdiagramforyourinvestment.MeasuretheGlobalexindexonthehorizontalaxisandtheportfoliorateofreturnontheverticalaxis.Whatistheslopeofthepayofflinetotherightofanindexvalueof104?

b.Ifyouthinkthatthereisaprobabilityof.5thattheGlobalexindexayearfromnowwillbeup12%,aprobabilityof.25thatitwillbeup40%,andaprobabilityof.25thatitwillbedown20%,whatistheprobabilitydistributionofyourportfoliorateofreturn?

SOLUTION:

a.

Toinsurethatyouwillatleastgetbackyouroriginal$1million,youneedtoinvest

inT-bills.

Youcanbuy

options.

Theslopeofthepayofflinetotherightofanindexvalueof104is4807.69,asseenfromthegraph:

b.

Put-CallParity

4.

a.Showhowonecanreplicateaone-yearpurediscountbondwithafacevalueof$100usingashareofstock,aputandacall.

b.SupposethatS=$100,P=$10,andC=$15.Whatmustbetheone-yearinterestrate?

c.Showthatiftheone-yearrisk-freeinterestrateislowerthaninyouranswertopartb,therewouldbeanarbitrageopportunity.（Hint:

Thepriceofthepurediscountbondwouldbetoohigh）.

SOLUTION:

a.Toreplicateaone-yearpurediscountbondwithafacevalueof$100,buyashareofstock,andaEuropeanputwithexerciseprice$100,andsellaEuropeancallwithanexerciseprice$100.

b.S=$100,P=$10,andC=$15.

E/（1+r）=S+P-C

$100/（1+r）=$100+$10-$15=$95

r=100/95-1=.053or5.3%

c.Ifr=4%,thenonecouldmakerisk-freearbitrageprofitsbyborrowingat4%andinvestinginsynthetic1-yearpurediscountbondsconsistingofashareofstock,aEuropeanputwithexerciseprice$100,andashortpositioninaEuropeancallwithanexerciseprice$100.Thesyntheticbondwouldcost$95andpayoff$100atmaturityin1year.Theprincipalandinterestonthe$95itcoststobuythissyntheticbondwouldbe$95x1.04=$98.8.Thustherewouldbeapurearbitrageprofitof$1.20perbondayearfromnowwithzeroinitialoutlayoffunds.

5.A90-dayEuropeancalloptiononashareofthestockofToshiroCorporationiscurrentlytradingat2,000yenwhereasthecurrentpriceoftheshareitselfis2,400yen.90-dayzero-couponsecuritiesissuedbythegovernmentofJapanaresellingfor9,855yenper10,000yenfacevalue.Inferthepriceofa90-dayEuropeanputoptiononthisstockifboththecallandputhaveacommonexercisepriceof500yen.

SOLUTION:

Usingtheexpressionforput-callparity,P=-S+E/（1+r）T+C

Sistheshareprice,Pisthepriceoftheput,CisthepriceofthecallandEisthecommonexerciseprice.

Sincegovernmentbondsaresellingat.9855per1yenoffacevalue,thisisthediscountfactorforcomputingthePVoftheexerciseprice.Thereisnoneedtocomputetherisklessrate,r.

Substitutingintheparityequationweget:

P=-2,400+500x.9855+2,000=92.75yen

6.GordonGekkohasassembledaportfolioconsistingoften90-dayUSTreasurybills,eachhavingafacevalueof$1,000andacurrentpriceof$990.10,and20090-dayEuropeancalloptions,eachwrittenonashareofParamountstockandhavinganexercisepriceof$50.00.Gekkoisofferingtotradeyouthisportfoliofor300sharesofParamountstock,whichiscurrentlyvaluedat$215.00ashare.If90-dayEuropeanputoptionsonParamountstockwitha$50.00exercisepricearecurrentlyvaluedat$25.00,

a.InferthevalueofthecallsinGekko’sportfolio.

b.DeterminewhetheryoushouldacceptGekko’soffer.

SOLUTION:

a.Usingput-callparity,thecurrentpriceofacallisfoundtobeapproximately$190.50asfollows:

C=S-E/（1+r）T+P=$215-$50x.9901+$25=$190.495

b.ThevalueofGekko’sportfoliois10x$990.10+200x$190.495=$48,000

Butthevalueof300sharesis$64,500.WeshouldrejectGekko’soffer.

7.

ThestockofKakkonen,Ltd.,ahottunadistributor,currentlylistsfor$500.00ashare,whereasone-yearEuropeancalloptionsonthisstock,withanexercisepriceof$200.00,sellfor$400.00andEuropeanputoptionswithasimilarexpirationdateandexercisepricesellfor$84.57.

a.Infertheyieldonaone-year,zero-couponU.S.governmentbondsoldtoday.

b.Ifthisyieldisactuallyat9%,constructaprofitabletradetoexploitthepotentialforarbitrage.

SOLUTION:

a.Usingput-callparity,wecaninfertherisklessyieldtobeapproximately8.36%asfollows:

Aportfolioconsistingofashareofthestock,aput,andashortpositioninacallisequivalenttoa1-yearT-billwithafacevalueofE.ThereforethepriceofsuchaT-billwouldhavetobe$184.57:

E/（1+r）T=S+P-C=$500.00+$84.57-$400.00=$184.57

1+r=200/184.57=1.0836

r=.0836or8.36%

b.TherearemanywaystoexploittheviolationoftheLawofOnePricetomakearbitrageprofits.Sincetherisk-freeinterestrateis9%,andtheimpliedinterestrateonthereplicatingportfoliois8.36%,wecouldgoshortthereplicatingportfolioandinvesttheproceedsinT-bills.Forexample,atcurrentprices,short-sella“unit”portfolio,whichconsistsoflongpositionsinoneputandoneshareandwritingonecall,toearnimmediaterevenueof$184.57.Theportfolioyousoldshortrequirespaymentof$200oneyearfromnow.Ifyouinvestthe$184.57inone-yearT-billsyouwillhave1.09x$184.57=$201.18ayearfromnow.Thusyouwillearnarisk-freearbitrageprofitof$1.18withnooutlayofyourownmoney.

Two-StateOptionPricing

8.Derivetheformulaforthepriceofaputoptionusingthetwo-statemodel.

SOLUTION:

Topricetheputoption,wecreateasyntheticoptionbysellingshortafraction（denotethefraction“a”）andlend$binrisk-freeasset.DenotethepriceofthestockS,thepriceoftheputoptionP,thestockpricewhenthenextperiodisan“up”tobeuS,thestockpricewhenthenextperiodisa“down”tobedS,thepayoffsoftheputoptionineachstatePuandPd,andtheris