CFA二級(jí)基礎(chǔ)段衍生（打印版）

LevelIISessionContentWeightingsEthics&ProfessionalSession1-2Session3QuantitativeMethodsEconomicAnalysisSession4Session5-6Session7-8Session9-11Session12-13FinancialStatementAnalysisCorporateFinanceEquityAnalysisFixedIncomeAnalysisDerivativeInvestmentsAlternativeInvestments5-10StudySessionSessionSessionPortfolioManagementSS14DerivativeInstruments——andStrategies?R37PricingandofCommitments?R38ofContingentClaims37PricingandValuationofForwardCommitments1.?????PrincipleofArbitrage-freePricingEquityandFuturesContractsandFuturesContracts(FRA)Fixed-IncomeandFuturesContractsCurrencyContracts23..T-bondFutures???ContractsCurrencyContractsEquityContracts??Aisagreementbetweentwopartiesinwhichonethetobuyfromtheothertheunderlyingassetorotherderivative,afutureapriceestablishedtheofthecontract.Thepartytothecontractthattobuythefinancialorphysicalassethasalongpositionandiscalledthelong.Thepartytothecontractthattosell/delivertheassethasashortpositionandiscalledtheshort.Priceand?Thepriceisthepredeterminedpriceinthecontractthatthelongshouldtotheshorttobuytheunderlyingassetthesettlement??valueistobothpartiesinitiationno-arbitrageprinciple:thereshouldnotbearisklessprofittobegainedbyacombinationofacontractpositionwithpositioninotherasset.zzzassetsorportfolioswithidenticalfuturecashflows,offutureevents,shouldsamepriceTheportfolioshouldyieldtherisk-freeofreturn,ifitcertainGeneralformula:=Sh(1+RTGenericPricing:No-ArbitragePrinciple?Pricingaistheprocessofdeterminingtheno-arbitragepricethatwillthevalueofthecontractbetobothsidestheinitiationofthecontractzprice=pricethatwouldnotpermitprofitablerisklessarbitrageinfrictionless??FP=SηCarryingιCarryingBenefitsameansdeterminingtheofthecontracttothelong(ortheshort)sometimeduringthelifeofthecontract.???Cash-and-CarryArbitrageWhentheContractisOverpricedzIfh(1+RTinitiationsettlementDelivertheunderlyingtothelongGetfromthelongShortacontractBorrow0therisk-freeUsethemoneytobuytheunderlyingbondtheloanamountofSh(1+RTProfit=Sh(1+RfTReverseCash-and-CarryArbitragewhentheContractisUnder-pricedzIf<Sh(1+RTinitiationsettlementLongaforwardcontractShortselltheunderlyingbondgetStheshortFPgettheunderlyingbondCloseouttheshortpositionbydeliveringthebondS0therisk-freeReceiveinvestmentproceedsSh(1+R)TProfit=Sh(1+R)T-FPT-bill(zero-couponzbuyaT-billtodaythespotprice(S)andshortaT-monthT-billcontracttheprice)36?u?5I7zvalueoflongpositioninitiation(t=0),duringthecontractlife(t=t),andexpiration(t=T)TimeContractt=0becausethecontractispricedtopreventarbitrageFP?Rf)T?tSt?t=tt=T-FPEquity?contractsonadividend-payingzzPrice:(S?)u?RT?Rf)TtS??FT?-FT?1?RfTtaFT??FTo=??Example?Assumingaforwardcontractwithdaysuntilmaturityonastock,thestockpriceandexpectedtodividend$0.3days,anddays.Theriskfreerate4%.Calculatetheno-arbitrageforwardprice.?CorrectAnswer:S$?1-uExample?Afterdays,thestockpricechangedtoCalculatethevaluationtheforwardcontract.?CorrectAnswer:zonlyonedividendremainingdays)beforethecontractmaturesdays)asshownso:?$$)--Example???aaAzT?PVFT??FTT?100.05?100.20?1?0.0030?0.149925?per?100parvalue?z??EquityIndex?contractsequityindexzzzContinuouslycompoundedrisk-freeR=ln(1+R)Continuouslycompoundeddividendyield:δcPrice:Sue(Rc?GcT0§S·§·long¨??¨??z¨?ec1?ec1Example?AssumingaforwardcontractontheDowJonesIndexwithdays.Currently,theDowJonesIndexandthecontinuousdividendyield2%.Thecontinuouslycompoundedriskfreerate3.2%.Calculatetheno-arbitragepricetheforwardcontract.=21,000ue(0.032-0.02)u(100/365)=21,069.1547?Afterdays,theDowJonesIndexKeeptheriskfreerateanddividendyieldsameasbefore.Calculatethevaluationtheforwardcontract.20,05021,069.1547eu(longposition)=-=-1,000.481euonCouponBonds?CouponzzzSimilartodividend-payingstocks,butthecashflowscouponsPrice:(S0?0)u?RfT(S?)??RfT?ttT[FT??FT]0Example?AssumingaforwardcontractwithdaysonaUStreasurybill.TheUStreasurybillhasacouponrate,theprice(includingaccruedinterest)andwillcouponpaymentdays.Theriskfreerate4%.Calculatetheforwardprice.?CorrectAnswer:$1000u0.05C$25$24.75942$1.0425.00PVC0zTheforwardpricethecontracttherefore:FP(onaincome=($1,100-$24.7594)u150/365=1,092.7122Currency?Price:coveredRateParity(IRP)?D)S0u?R)TFPandS0quotedinDperunitofF(i.e.,D/F)??S?R)??R)T?tIfgiventhecontinuousinterFPS0ue(Rc?Rc)T§?StRcu(T?t)·1§?FPRcu(T?t)·1long¨??¨?eeExample?Considerthefollowing:TheU.S.risk-freerare6percent,theSwissrisk-freerate4percent,andthespotexchangeratebetweentheUnitedandSwitzerlandzzzCalculatethecontinuouslycompoundedU.S.andSwissrisk-freerates.Calculatethepriceatwhichyoucouldenterintoaforwardcontractthatexpiresdays.Calculatethetheforwardpositiondaysintothecontract.AssumethatthespotrateCurrencyForwardContracts?Answer:zrfc=zS0=zT=90/365r=ln(1.06)=0.0583F(0,T)=($0.666uee)=$0.6698zt=zT=90/365zt=25/365zT-t=65/365t(0,T)=($0.65ue)-($0.6698ue)=zThethecontractperSwissfrancAgreements(FRAs)?A(FRA)isaan(LIBOR).zzzThelongpositioncanbeviewedtherightandtheobligationtoborrowtheinthefuture;Theshortpositioncanbeviewedtherightandtheobligationtolendtheinthefuture.Noloanisactuallymade,andFRAssettledincashcontractexpiration.?a1h4FRAA1h4FRAiszzzzacontractin1month,andtheunderlyingloanissettledin4months,witha3-monthnotionalloanperiod.Theunderlyingis90-dayLIBORin30fromAgreements(FRAs)?LIBOR:LondonInterbankRate.zzzannualizedbasedona360-dayadd-onoftenusedaforfloatingdollar-denominatedloansworldwide.zpublisheddailybytheBritishAssociation?Euribor:InterbankinpublishedbyEuropeanCentralBank.–?LIBOR,FRAsδ?ε??φsettleincash,butnoactualloanismadethesettlementz????φ99Ifthetheexpirationisabovethespecifiedcontractthelongwillreceivecashpaymentfromtheshort;Ifthetheexpirationisbelowthecontracttheshortwillreceivecashpaymentfromthelongz????aadaysoo?rate????????????adayso?1????????Example?Indays,aUKcompanyexpectstoabankdepositforaperioddaysat90-dayLiborsetdaysfromThecompanyconcernedaboutapossibledecreaseinterestrates.ItsfinancialadvisersuggeststhatnegotiateatTimea1h4FRA,aninstrumentthatexpiresdaysandbasedon90-dayThecompanyentersintoanotionalamount1h4receive-fixedFRAthatadvancedset,advancedsettled.TheappropriatediscountratefortheFRAsettlementcashflows0.40%.Afterdays,90-dayLiborBritishpounds0.55%.IftheFRAwasinitiallypricedat0.60%,thepaymentreceivedsettlewillbeclosestto:A.–￡2,448.75.C.Example?Solution:Bcorrect.Inthisexample,m=(numberdaysthedeposit),tm=0/360(fractionyearuntildepositmaturesobservedattheFRA9expirationdate),andh=(numberdaysinitiallytheFRA).Thesettlementamountthe1h4FRAathforreceive-fixedNA{[FRA(0,h,m)–L(m)]t}/[1+D(m)t]=–+=BecausetheFRAinvolvespayingfloating,benefitedfromadeclinerates.FRAPricing?ThepriceinanFRAistheno-arbitrage(FR)zIfknown,ThetheofL(m)/m/n+n)/m+n(1?mum/?FRun/?m?nu(m?)/Example?Calculatepriceah4FRA.30-dayLIBORand20-dayLIBOR3.9%.Answer:1?zzzTheactual30-dayrate(Period):R(30)=0.03h=Theactual120-dayrate(Period):R(120)=0.039h=Theactual90-dayforwardratedaysfromnow(period):(1+R(120))/(1+R(30))-1=/-zTheannualizedforwardrate,whichthepricetheFRA,:R=0.015h==4.2%.Example?Supposeweenteredareceive-floating6h9FRAatarate0.86%,withnotionalamountC$10,000,000atTimeThesix-monthspotCanadiandollar(C$)Liborwas0.628%,andthenine-monthC$Liborwas0.712%.Also,assumethe6h9FRAratequotedthemarketat0.86%.Afterdayshavepassed,thethree-monthC$Liborandthesix-monthC$Libor1.35%,whichwewilluseasthediscountratetodeterminetheatg.haveh=andm=90.AssumingtheappropriatediscountrateC$thetheoriginalreceive-floating6h9FRAwillbeclosestto:A.C$14,500.?B.C$14,625.C.C$14,651.Example8-Solution?CL=L=L+=L=0L==L–==L+m–=L=g–=–==+g+m–th+m–g+g–h–g]–m,=++–+–9–=++–1==Example8-Solution?Solution:Therefore,(0,h,m)==–0.0086)(90/360)]/[1+=Again,floatingratesroseduringthisperiod;hence,theFRAenjoyedagain.NoticethattheFRAraterosebyroughlybps–and1bpfor90-daymoneyandanotionalamountThus,wecanalsoestimatetheterminalas10hh=Aswithallfixed-incomestrategies,understandingtheabasispointoftenhelpfulwhenestimatingandlossesandmanagingtherisksFRAs.1.?????PrincipleofArbitrage-freePricingEquityandFuturesContractsandFuturesContracts(FRA)Fixed-IncomeandFuturesContractsCurrencyContracts23..T-bondFutures???ContractsCurrencyContractsEquityContractsFutures??Thevalueofafuturescontractiscontractinception.Futurescontractstothevalueaftermarkingtomarketisresetto.?Betweenthetimeswhichthecontractistothevaluecanbedifferent.zV(long)=currentfuturesprice?futurespricethemark-to-markettime.?Anotherviewoffutures:settlepreviousfutures,andthenopenanothernewfutureswithsamedateofT-bondFutures??Underlying:Hypothetical30treasurybondwith6%couponBondcanbedeliverable:$100,000parvalueT-bondswithcouponbutwithamaturityof15??Thequotesinpointsand32nds:Apricequoteof95-18isequalto95.5625andadollarquoteof$95,562.50Theshorthasadeliveryoptiontochoosewhichbondtobondisgivenafactor(CF),whichmeansaspecificbondisequivalenttoCFbondunderlyinginfuturescontract.??Theshortdesignateswhichbondhewilldelivercheapest-to-deliverbond).aspecificBondA:1uAQuotedfuturesprice????BondpriceisusuallyquotedascleanpricezCleanprice=fullprice-accruedthefuturespricecanbewrittenas(S?)u?RTSu?RT?IfS0isgivenbycleanprice(quotedfrice)f(S?)u?R)T?IfthefuturespriceisquotedascleanpricefzNotedtAII+RT)u?R)??AIT?Thequotedfuturespriceiswithfactor1QFPa?S+AI?u(1?R)T?AIT?FVCou?fCFExample?Euro-bundfutureshaveacontractandtheunderlyingconsistslong-termGermandebtinstrumentswith8.5toyearsTheyaretradedontheEurex.SupposetheunderlyingGermanbundquotedatandhasaccruedinterest(one-halfamonthsincelastcoupon).Theeuro-bundfuturescontractmaturesonemonth.contractexpiration,theunderlyingbundwillhaveaccruedinteresttherearenocouponpaymentsdueuntilafterthefuturescontractexpires,andthecurrentone-monthrisk-freerate0.1%.Theconversionfactor0Inthiscase,wehaveT=1/12,CF(T)=B+Y)=1FVCI=AI0==AIT==andr=0.1%.Theequilibriumeuro-bundfuturespricebasedonthecarryarbitragemodelwillbeclosestto:A.B.C.Example-Solution?Solution:Bcorrect.Thecarryarbitragemodelforforwardsandfuturessimplythefuturetheunderlyingwithadjustmentsforuniquecarryfeatures.Withbondfutures,theuniquefeaturesincludetheconversionaccruedinterest,andanycouponpayments.Thus,theequilibriumeuro-bundfuturespricecanbefoundusingthecarryarbitragemodelwhichF(T)=FV–AIT–FVCIorQF(T)=[1/CF(T)]{FV+Y)+AI]–AI–FVCI}Thus,wehaveQF(T)=++–0.25–Inequilibrium,theeuro-bundfuturespriceshouldbeapproximatelybasedonthecarryarbitragemodel.=€Example?identifiesanarbitrageopportunityrelatingtoafixed-incomefuturescontractanditsunderlyingbond.CurrentdataonthefuturescontractandunderlyingbondarepresentedExhibit.Thecurrentannualcompoundedrisk-freerate0.30%.Example??B0??????????????????a–=00=1.?????PrincipleofArbitrage-freePricingEquityandFuturesContractsandFuturesContracts(FRA)Fixed-IncomeandFuturesContractsCurrencyContracts23..T-bondFutures???ContractsCurrencyContractsEquityContractsPricingaplainswap??AplainvanillaisinwhichonepartyafixedandtheotherafloatingzPricingaplainvanillameanscalculatingthefixed()thatthecontractvalueinitiation.Sinceafloating-ratebondhasavalueequaltoparvalueinitiation,whatwilldoistofindabondwithavalueequaltotheparvalueinitiation.DenoteCthecouponofthen-periodfixed-ratebond,Pricingaplainswap??1=ChB1+ChB2+ChB3+……+ChBn+1hBnAndthencangettheCas:1?CB?B??RecallthatBnisthediscountwhichispresentvalueinnperiods.It’simportanttonotethattheanswerCaperiodicandmustannualizeittogettheannualExample?Calculatetheswaprateofa1-yearquarterly-payplainvanillaswap.day)=3.5%;R(360-day)=4%.?Answer:zStep1:Calculatethediscountfactors:hhh=0h0zC=(1?==?0.9615)/(0.9938+0.9852+0.9744+0.9615)ExamplezStepCalculatetheannualizedswaprate:swaprate=0.98%h=swapratecanbeviewedspotrates.Soeverytimeyouaswapratealwayscheckwithintherangespotrates.example,getandthiswithinrange2.5%and4%.Anothertrickthattheswaprateusuallyveryclosetothelastspotratehere).aplainswap?Thevalueofareceive-fixed,pay-floatingissimplythevalueofbuyingafixed-ratebondandissuingafloating-ratebond.fixedCompanyXCompanyYfloatingzNotes:thevalueafloatingbondwillbethenotionalamountitsperiodicsettlementwhenpaymentissettothe(floating).?Thevaluationformula:(X)B-BY)B-BVtT[F?rate??F?rate]0Example?Calculatethetheplainswappay-fixedpreviousexampleafterdays.notionalprincipal$1million.AssumeafterdaysspotratesR(60-day)=3%;R(150-day)=3.5%;R(240-day)=4%;R(330-day)=4.5%.?AnswerzCalculatethenewdiscountfactorsdayslater:=1/(1+3%h60/360)=0.9950B2=1/(1+3.5%h150/360)==1/(1+4%h240/360)=0.9740;B4=1/(1+4.5%h330/360)=;00zCalculatethethefixed-ratebond:P(fixed)=0.98%h(0.9950+0.9856+0.9740+0.9604)+1h0.9604=ExamplezStep3:Calculatethethefloating-ratebond:0dayswhenthefirstcomes,floating-ratebondpricewillThefirstknownatinitiation:62.5%h90/360=0.00625.Sowegetthefloating-ratebondpriceas:P(floating)=h=zStep4:Calculatetheswaptothepay-fixedside:V=[P(floating)?P(fixed)]hnotionalprincipal=hmillion=Example?Q1:Supposeyouarepricingafive-yearLibor-basedinterestrateswap.Theestimatedpresentfactorsareasfollows.Calculatethefixedswaprate.PresentFactor0.990099123450.9778760.9651360.9515290.937467?CorrectAnswer:?B1?2?11?C0.990099?0.977876?0.965136?0.951529?zBecausea5-yearannually-payplainswap,Swaprateequal1.2968%.Example?Q2:Supposetwoyearsagoweentereda7-yearreceive-fixedLibor-basedinterestrateswapwithannualresetsdayaccount).Thefixedratetheswapcontractenteredtwoyearsagowas2%.ThepresentfactorsaresameasaspresentedCalculatetheforthepartytheswapreceivingthefixedrate.?CorrectAnswer:ztheswapperdollarnotionalV?2%??u?0.990099?0.977876?0.965136?0.951529??zThus,thetheswapPricingacurrencyswap??Consideraswapinvolvingtwocurrencies,theUSdollarandtheEuroTheexchangeratenowAswedoaplainswap,wecangetthefixedratethatwillthefixed$paymentsequalto$1,andthefixedratethatwillthefixed€paymentsequal€1.Forexample,zIftheUStermstructureis:R(90-day)=5.2%,R(180-day)=5.4%,R(270-day)=5.55%,R(360-day)=5.7%,wecangetthefixedrate5.56%.zIftheEurotermstructureis:R(90-day)=3.45%,R(180-day)=3.58%,R(270-day)=3.7%,R(360-day)=3.75%,wecangetthefixedrate3.68%.?OurcurrencyswapinvolvingdollarsforEuroswouldhaveafixedrate5.56%dollarsandEuros.Thenotionalprincipalwouldbeand0.8.Therearefourwaystoconstructtheswap:€Pricingacurrencyswap?tothezzzz1:dollarfixed5.56%andreceiveEurofixed3.68%.2:dollarfixed5.56%andreceiveEurofloating.3:dollarfloatingandreceiveEurofixed3.68%.4:dollarfloatingandreceiveEurofloating.9Inthe4(floatingforfloating),thereisnopricingproblembecausethereisnofixedshouldonlysetthenotionalprincipalto€0.8forevery1$.Example?Somedaystermstructurescountrieswillchange,theexchangewillalsobedifferent.cancalculatefixed-andfloating-ratebondpricesbothcurrencies,andthenwillgettheswapjustasyoudoaplainswap.GBP6%CompanyACompanyBUSD5%Ais:swap)USD-(S0uGBP)where:S0Example?Consideratwo-yearcurrencyswapwithsemiannualpayments.ThedomesticcurrencytheU.S.andtheforeigncurrencytheU.K.pound.Thecurrentexchangerateperpound.L=L=L=L=L=L=L=L=Thecomparableset?ratesarezA.Calculatetheannualizedfixedratesfordollarsandpounds:Example?AnswerforA:B=0=B=0=B=0=B=0=is1-=+++Example?======?==z999?$?$?$Example?AnswerforB:zThenewdollarandpoundfactorsforanddaysasfollows:B=B=B=B=====ExamplezInterms$payments:9Thepresenttheremainingfixedpaymentsplusthenotionalprincipal++++=Thepresentthefloatingpaymentsplushypothetical1notionalprincipaldiscountedbackdays+=9$[ExamplezInterms￡payments:9Thepresenttheremainingfixedpaymentsplusthenotionalprincipal++++=9Convertthisamounttotheequivalentnotionalprincipalandconverttodollarsatthecurrentexchangerate:1/1.41**=Example9Thepresentthefloatingpaymentsplushypothetical1notionalprincipal+=Converttodollarsatthecurrentexchangerate1/1.41**=zThemarketvaluesbasedonnotionalprincipalareasfollows:￡1999￡fixedandreceive$fixed==-￡andreceive$fixed==-￡andreceive$==-9￡fixedandreceive$floating==-Example?AUScompanyneededborrowmillionAustraliandollars(A$)foroneyearforAustraliansubsidiary.ThecompanydecidedtoborrowUSdollars(US$)anamountequivalentA$100millionbyissuingUS-denominatedbonds.Thecompanyenteredintoaone-yearcurrencyswapwithaswapTheswapusesquarterlyresetdaycount)andexchangenotionalamountsatinitiationandat?theexpiration,theUScompanypaysthenotionalamountAustraliandollarsandreceivesfromthedealerthenotionalamountUSdollars.ThefixedrateswereforAustraliandollarsand0.2497%forUSdollars.thenotionalamountUSdollarswaswithaspotexchangerateA$1.14forUS$1.Example?Assumedayshavepassedandweobservethefollowingmarketinformation:A$SpotInterestRates(%)US$SpotInterestRates(%)DaystoMaturityPresent(A$1)Present(US$1)30Sum:Sum:Example?ThecurrenttotheswapdealerA$thecurrencyswapenteredintodaysagowillbeclosestto:A.–A$13,557,000.C.Example?Q1CorrectAnswer:C.zInitialCashFlowsExchanged=USFirmSwapDealerSwapDealerzQuarterlyCashFlowsExchanged=USFirmzCashFlowsExchangedUSFirmSwapDealerExample?Q1CorrectAnswer:C.zTheUSfirmissuesabondandentersaswapwiththeswapTheinitialexchangerategivenasA$1.14forUS$1.TheswapdealerreceivingquarterlyinterestpaymentscurrencyA$.zAfterdaysthenewexchangerateA$1.13perUS$1,andthetermstructurehaschangedbothmarkets.ThisthetheswapthepartyreceivinginterestpaymentsAustraliandollars,whichtheswapExamplezThePVthedealer’sincomingcashflowsA$,effectivelyalongpositionA$bond.m?[(2.7695%/4)?(+=A$101,350,248.39+zThePVtheUSDoutflows(effectivelyashortUSDbondThePVthequarterlyinterestpaymentsandterminalpaymentcalculatedusingthenewtermstructureandconvertedinto9?(USD)?[(0.2497%/4)?(0.999584+0.998668+0.998253+0.998336)+=A$z-=A$?Example?ThecurrentUSDtotheUSfirmthecurrencyswapenteredintodaysagowillbeclosestto:A.–$1,898,400.C.Example?Q2CorrectAnswer:B.zThetotheUSfirmThisrepresentsthetothefirmmakinginterestpaymentsinCurrencya(A$)zVfirm=-A$2,145,203whichconvertedUSD–h=–$1,898,410zNotethattheUScompanyissues(short)abondUSDtheirhomemarketandusesaswapeffectivelyconvertbondissue.Understandingtheswapastwobonds,theUSfirmlongaUSDbondandshortabondA$.TheswapoffsetstheUSUSDbondissue(short).TheswapallowstheUSfirmtoA$interestpaymentstotheswaportoeffectivelyissueabondA$.Pricinganequityswap??Therethreetypesofequityswaps:(1)andreceiveequityreturn;(2)floatingandreceiveequityreturn;(3)oneequityreturnandreceiveanotherequity.onlyneedtopricethetypebecausetherenofixedintheothertwo.thesameformulafortheplaintogettheperiodicofequity1?CB?B?"?B?isthiscase?canabondperiodiccouponofCforaindexwiththenotionalamountequaltotheparvaluethebond,becausethebondvalueinceptionisequaltoanequityswap?$1ExampleAnswer:?Step1:Calculatethenewdiscountfactorsdayslater:B1=1/(1+3%h=B3=1/(1+4%h=;;B2=1/(1+3.5%h=0B4=1/(1+4.5%h=0???Step2:Calculatethethefixed-ratebond:P(fixed)=0.98%h(0.9950+0.9856+0.9740+0.9604)+1h0.9604=StepCalculatethetheindexinvestment:P(index)=1h=1.1Step4:Calculatetheswaptothefixed-ratepayer:V=[P(index)?P(fixed)]hnotionalprincipal=(1.1-0.998767)hmillion=Swaption?Aswaptionisoptiontointoa.willfocusontheplainvanillaswaption.Thenotationswaptionsissimilartoaswaptionmaturesin2andgivestheholderrighttointoa3-theendofthesecondisa2h5swaption.??Apayerswaptionisoptiontointoathefixed-ratezSoaaazaAreceiverswaptionoptiontoathefixed-ratereceiver(thefloating-rateincreases,receivervaluegodown.areceiverswaptionisequivalenttoacalloptiononacouponbond.??aplainvanillaswaptionisequivalenttovaluingunderlyingifthevalueislowerthanexpiration,thevalueofaplainvanillaswaptionwillbeExample:Aquarterly-payswaptionwiththeof3.84%andthenotionalprincipal$1millioncomestoitsexpirationThespotR(90-day)=2.5%;R(180-day)=3%;R(270-day)=3.5%;R(360-day)=4%.ThecurrentonisCalculatethevalueoftheswaption.Answer:?Step1:Calculatethediscountfactors:B1=1/(1+2.5%h90/360)=0.9938;B2=1/(1+3%h180/360)=0.9852B3=1/(1+3.5%h270/360)=0.9744B4=1/(1+4%h360/360)=0.9615;?Step2:Calculatethenetcashsavingsateachpaymentdate:(3.92%3.84%)h90/360hmillion=$200??Step3:Calculatethepresentvalueofthesavings:$200h(0.9938+0.9852+0.9744+0.9615)=$783TheownerswaptioncantheswaptionandaOrhecanagreetoreceivequarterlypaymentsof$200counterpart.Hecanalsotoreceive$783todaytoterminatethecontract.?Iftheis4%intheswaption,thevalueoftheunderlyingis$783andtheswaptionholderwillnottheswaption40ValuationofContingentClaims1.BinomialModel——TheExpectationsApproach???Theone–periodbinomialstockmodelThetwo–periodbinomialstockmodelbinomialmodel2345....Black-Scholes-MertonModelOptionandImpliedVolatilityBinomialModel——TheNo-arbitrageApproachBlackOptionModel???OptionOnFuturesOptionSwaptionsPut-callparityEuropeanoptions??Afiduciarycallisaportfolioconsistingof:zAlongpositioninaEuropeancalloptionwithpriceofXthatmaturitiesinTonastock.zAlongpositioninapure-discountrisklessbondthatXinTTheafiduciarycallisthecostofthecall)plusthecostofthebond(thepresentvalueofThetoafiduciarycallwillbeXifthecallisout-of-the-moneyandTifthecallisin-the-money,showninthefollowing:SXSCallout-oforat-the-money)Callisin-the-money)LongcallLongbond0XXST-XXSTPut-callparityEuropeanoptions?Aputaof:zAlongpositioninaEuropeanputoptionwithanexercisepriceofXthatmaturitiesinTyearsonastock.zAlongpositionintheunderlyingstock.??Thecostofaprotectiveputisthecostoftheput(P)plusthecostoftheThepayofftoaprotectiveputisXiftheputisin-the-moneyandSTiftheputisout-of-the-money.asshowninthefollowing:最新資料通知微信：xuebajun888sS<XS≥X(putisin-the-money)(putisout-oforat-the-money)LongputLongstockX-TT0TXSTsyntheticinstruments?Theresyntheticinstruments:XCP?S?zzzzAsyntheticEuropeancalloption:?RfTsyntheticcall=put+?bondXP0??S0AsyntheticEuropeanputoption:?Rf)Tsynthetic=call+?X0?S0?0Asyntheticpure-discountrisk-lessbond:?RfTsyntheticbond=put+?callXAsyntheticstockposition:S00??P?R)synthetic=call+?putsyntheticinstruments?Theretworeasonsmightwantstosyntheticpositionsinthesecurities.zpriceoptionsbyusingcombinationsoftheotherinstrumentswithknownprices.zearnarbitrageprofitsbyexploitingrelativemispricingamongthefoursecurities.Ifput-callparitydoesn’thold,arbitrageprofitisavailable.Exploitviolationsput-callparity?Aswithallarbitragetrades,wanttolowsellput-callparitydoesn’tholdcostafiduciarycalldoesnotequalcostaprotectiveput),you(golongin)theunderpricedpositionandsell(goshort)theoverpricedposition.?Example:Exploitviolationsput-callparityz90-dayEuropeancallputoptionswithapricepricedatandTheunderlyingpricedatandmakesnocashduringtheoptions.Therisk-free5%.Calculatetheno-arbitragepricecalloption,andillustratehowearnanarbitrageExploitviolationsput-callparityAnswer:??C0=P0+S0?X/(1+RT=+$48?=Sincethecalloverpricedzweshouldsellthecallfor$7.50andbuythesyntheticcallzsyntheticcall,buyforbuytheunderlyingforandshort)a90-dayzero-couponbondwithafacezThetransactionwillgenerateanarbitrageOne-periodbinomialmodel??Abinomialmodelisbasedontheideathat,overthenextperiod,somevaluewillchangetooneoftwopossiblevalues(binomial).constructabinomialmodel,needtoknowthebeginningassetvalue,thesizeofthetwopossiblechanges,andtheprobabilitiesofeachofthesechangesoccurring.offbyhavingonlyonebinomialperiod,whichmeansthattheunderlyingpricemovestotwonewpricesoptionexpiration.let0bethepriceoftheunderlyingstockOneperiodthestockpricecanmoveuptoordownto.thenidentifyau,theupmoveonthestockanddthedownmove.Thus,uandd.furtherassumethatu=1/d.?S0uS??=SdExpectationapproach??Risk-neutralprobabilityofupmoveisπu;Risk-neutralprobabilityofdownmoveisπ=1-π:1?R?du?dSuwithacalloption.thegoesupto,thecalloptionwillbeworthC.downto,thecalloptionwillbeworthC.knowthatthevalueofacalloptionwillbeintrinsicvalueonexpirationThusweget:C=Max(0,;C=Max(0,1cC??SdC?ou??u?RfTC++--h?ratio?=?SExample?Calculatethetodaya1-yearcalloptiononthestockwiththepriceThepricethestockandthesizeanup-moveTherisk-freerate7%.?Answer:zCalculatetheparameters:9u=1.25;d=1/u=0.8;S=20hS=20h0.8=169C=Max?=5;C=Max?=0zCalculaterisk-neutralprobabilities,πuandπd=1?πu:9πu=(1+0.07??0.8)=0.6πd=1?πu=0.49zCalculatecalloptionprice:9C0=(5*0.6+0*0.4)/(1+7%)=2.80Example?Drawtheone-periodbinomialtree:=S=25;C=5(C=?)S=200.6.40=S=16;C=0todayyear1Example???Pricingaputoptionsimilartothatacall.TheonlydifferencethatP=MaxX?S)andP=MaxX?SExample:UsetheinformationthepreviousexamplecalculatethetodayaputonthesamestockwiththestrikepriceAnswer:P=Max?=P=Max?=4P=h0+0.4h=1.6/1.07=The–periodbinomialmodelS=C=Max(0,SS=uCSS====S(C=?)C=C=Max(0,S=SdCS=C=Max(0,S12Example?Calculatetodaya2-yearcalloptiononstockwithpriceThepricethestockandsizeanup-move1.25.Therisk-freerate7%.ExampleAnswer:?1:Calculatetheparameters:U=d=1/u=Su=h=Sd=h0.8=S=hh=S=S=hh0.8=C=Max?=C=Max?=2C=Max?=0?Calculaterisk-neutralprobabilities,πand?π:πu=(1+0.07??=0.6πd=?π=0.4Acallwithatwo-periodbinomialtree?Step3:thetwo-periodbinomialtree:S=S=31.25C