CFA二级基础段衍生（标准版）

二级培训项目IISessionContentWeightingsEthics&ProfessionalSessionMethodsEconomicAnalysisSession3Session4FinancialAnalysisCorporateFinanceSessionSessionSessionSessionEquityAnalysisFixedIncomeAnalysisDerivativeInvestmentsAlternativeInvestments5-10StudySessionSessionSessionPortfolioManagementSS14Derivative——Strategies••R37PricingandofCommitmentsR38ofContingentClaims37and1.FrameworkPrincipleofArbitrage-freePricingEquityandContractsandContracts(FRA)Fixed-IncomeandContractsCurrencyContracts23.T-bond.ContractsCurrencyContractsEquityContractsAispartiesinwhichonethetobuyfromtheothertheunderlyingassetorotherderivative,afutureapricetheofthecontract.Thepartytothecontractthattobuythefinancialorphysicalassethasalongpositionandiscalledthelong.Thepartytothecontractthattosell/delivertheassethasashortpositionandiscalledtheshort.andThepriceisthepriceinthecontractthatthelongshouldtotheshorttobuytheunderlyingassetthesettlementistobothpartiesinitiationprinciple:shouldnotbearisklessprofittobebyacombinationofacontractpositionwithpositioninotherasset.assetsorportfolioswithidenticalfuturecashflows,offutureevents,shouldsamepriceTheportfolioshouldyieldtherisk-freeofreturn,ifitformula:=S×(1+RT0fPricingaistheprocessofdeterminingtheno-arbitragepricethatwilltheofthecontractbetobothsidestheinitiationofthecontractprice=pricethatwouldnotpermitprofitablerisklessarbitrageinfrictionless＋Carrying－CarryingBenefitsameansdeterminingtheofthecontracttothelongtheshort)sometimeduringthelifeofthecontract.Cash-and-CarryWhentheisOverpricedIf×(1+RT0finitiationsettlementDelivertheunderlyingtothelongfromthelongShortacontractBorrow0therisk-freeUsethemoneytobuytheunderlyingbondtheloanamountofS×(1+RT0fProfit=S×(1+RT0fCash-and-CarrywhentheisUnder-pricedIf<S×T0fLongaShortsellS0shortFPCloseshortpositionS0S(T0f(T0f1T-billbuyaT-billthespotprice(S)andshortaT-monthT-billcontractthepriceSR)T0foflongpositioninitiation(t=0),duringthecontractandTimebecausethecontractispricedtoarbitrageFPRf)TtStT-FP1adividend-payingPrice:(S)R)Tf00VSlongttTtR)fFT-FTFTFT=t0long，t0T-t1tTRf1ExampleAssumingaforwardcontractwithdaysuntilmaturityonastock,thestockpriceandexpecteddividenddays,anddays.Theriskfreerate4%.Calculatetheno-arbitrageforwardprice.CorrectAnswer:S1200$1-1ExampleAfterdays,thestockpricechangedCalculatethetheforwardcontract.CorrectAnswer:onlyonedividendremainingdays)beforethecontractmaturesdays)asshownso:040120V?40$4035/3651$V)--4060/36511Exampleforwardwithexpirationaprice100.20aper-is¥100,000,000.forwardpriceis100.05.positionisA.-¥149,925.B.-¥150,000.C.-¥150,075.Solutions:AforwardpositioniscalculatedFTFTTVTtt0t2/12VT1t0¥,forwardpositionis0T1anequityContinuouslycompoundedrisk-freeR=lnR)fContinuouslycompoundeddividendyield:δcccfPrice:(R)TSef0StVlongc(Tt)R(Tt)ceef1ExampleAssumingaforwardcontractontheDowJonesIndexwithdays.theDowJonesIndexandthecontinuousdividendyield2%.Thecontinuouslycompoundedriskfreerate3.2%.Calculatetheno-arbitragepricetheforwardcontract.=21,000e-=21,069.1547Afterdays,theDowJonesIndexKeeptheriskfreerateanddividendyieldsameasbefore.Calculatethetheforwardcontract.20,05021,069.1547e0.032(25/365)V(longposition)=-=-1,000.48175e0.02(25/365)1Similartodividend-payingstocks,butthecashflowscoupons(S)RT00Price:f(S)ttR)TtfV[FTFT]longtTt01ExampleAssumingaforwardcontractwithdaysonaUStreasuryTheUStreasuryhasacouponrate,theprice(includingaccruedinterest)andwillcouponpaymentdays.Theriskfreerate4%.Calculatetheforwardprice.CorrectAnswer:$10000.05C$252$25.00PVC0$24.75941.0490/365Theforwardpricethecontracttherefore:FPa=-150/3651Price:Parity(IRP)D)TS0TF)SinDperunitF(i.e.,D/F)0StVlongR)TtR)TtFDIfgiventheccDFPSe(RR)TF0SFPVtlongcceR(Tt)eR(Tt)FD2ExampleConsiderthefollowing:TheU.S.risk-free6percent,therisk-freerate4percent,andthespotexchangeratebetweentheUnitedandSwitzerlandCalculatethecontinuouslycompoundedU.S.andrisk-freerates.Calculatethepriceatwhichyoucouldenterintoaforwardcontractthatexpiresdays.Calculatethetheforwardpositiondaysintothecontract.Assumethatthespotrate2CurrencyForwardContractsAnswer:rcr=S0=T=F=ee)=t=T=t=T-t=t=e)-e)=Thethecontractperfranc2(FRAs)Aisaan(LIBOR).Thelongpositioncanbetherightandtheobligationtoborrowtheinthefuture;Theshortpositioncanbetherightandtheobligationtolendtheinthefuture.Noloanisactuallymade,andFRAsincashcontracta1×4FRAA1×4FRAisacontractin1month,andtheunderlyingloanisin4months,witha3-monthnotionalloanperiod.Theunderlyingis90-dayLIBORin30from2(FRAs)LIBOR:Interbank.annualizedbasedona360-dayadd-onusedafloatingdollar-denominatedloansworldwide.publisheddailybytheBritishAssociationEuribor:inBank.2Pricingand–FRALIBOR,FRAs（续）交割：incash,butnoactualloanismadethesettlement定性分析：Ifthetheisabovethespecifiedcontractthelongwillreceivecashfromtheshort;Ifthetheisbelowthecontracttheshortwillreceivecashfromthelong定量分析rate360Notionalprincipal3602ExampleIndays,aUKcompanyexpectsadepositforaperioddaysat90-dayLiborsetdaysThecompanyconcernedaboutapossibledecreaseinterestrates.ItsfinancialadvisersuggeststhatnegotiateatTimea1×4FRA,aninstrumentthatexpiresdaysandbasedon90-dayThecompanyentersintoanotionalamount14receive-fixedFRAthatadvancedset,advancedsettled.The×appropriatediscountratefortheFRAsettlementcashflowsAfterdays,90-dayLiborBritishpounds0.55%.IftheFRAwaspricedatthepaymentreceivedsettlewillbeclosestto:A.–C.2ExampleSolution:Bcorrect.Inthisexample,m=(numberdaysthedeposit),=(fractionyearuntildepositmaturesobservedattheFRA9expirationdate),andh=(numberdaystheFRA).Thesettlementamountthe1×4FRAathforreceive-fixedNA{[FRA(0,h,m)–L+D]hmhm=–+=BecausetheFRAinvolvespayingbenefitedadeclinerates.2priceinanFRAisthe(FR)IfThetheof/m/n+n)/m+nLm/n/L(m)/mmn2ExampleCalculatepricea×4FRA.30-dayLIBORandLIBOR3.9%.Answer:1Theactualrate×=Theactualrate×=Theactualforwardratedaysnow(period):-1=/-Theannualizedforwardrate,whichthepricetheFRA,:RFRA×==4.2%.2ExampleSupposeweenteredareceive-floating6×9FRAataratewithnotionalamountatTimeThesix-monthspotCanadiandollar(C$)Liborwasandthenine-monthC$LiborwasAlso,assumethe6×9FRAratequotedthemarketatAfterdayshavepassed,thethree-monthC$Liborandthesix-monthC$Liborwhichwewilluseasthediscountratedeterminetheatg.haveh=andm=90.AssumingtheappropriatediscountrateC$thetheoriginalreceive-floating6×9FRAbeclosestto:A.C.3Example8-Solution:CisweL=L(180)=0.628%,L+=L(270)=00000.712%,FRA(0,180,90)=0.86%.After90=90),weL–=gL90)=1.25%L+m–=L(180)=1.35%.during9gthisperiod;FRAlikelygainedpositionisFirst,wenewTimegfairFRAinnewoldFRAnewFRATimeFRA(g,h–=FRA(90,90,90),ison90FRAin90inwhichunderlyingis90-dayis–=FRA(90,90,90)=+L+m–+L–]–1}/t,ggggmoninformationinthisexample,weFRA(90,90,90)=+L90(180+90–90)(180/360)]/[1+L90(180–90)(90/360)]–1}/(90/360).SubstitutinggiveninthiswefindFRA(90,90,90)=+0.0135(180/360)]/[1+0.0125(90/360)]–1}/(90/360)=[(1.00675/1.003125)–=0.0145,or1.45%.3Example8-Solution:Therefore,g(0,h,m)==–+=Again,ratesroseduringthisperiod;hence,theFRAenjoyedagain.NoticethattheFRAraterosebyroughlybps–and1bpformoneyandanotionalamountThus,wecanalsoestimatetheterminalas××=Aswithallfixed-incomestrategies,understandingtheapointoftenhelpfulwhenestimatingandlossesandmanagingtherisksFRAs.31.FrameworkPrincipleofArbitrage-freePricingEquityandContractsandContracts(FRA)Fixed-IncomeandContractsCurrencyContracts23.T-bond.ContractsCurrencyContractsEquityContracts3Theofafuturescontractiscontractinception.Futurescontractstotheaftermarkingtoisto.Betweenthetimeswhichthecontractistothecan.V(long)=currentfuturesprice−futurespricethemark-to-markettime.Anotherviewoffutures:settleandthenopenanothernewwith3Underlying:Hypothetical30treasurybondwith6%couponBondcanbedeliverable:$100,000parT-bondswithcouponbutwithamaturityof15Thequotesinpointsand32nds:Apricequoteof95-18isequalto95.5625andadollarquoteof$95,562.50Theshorthasadeliveryoptiontochoosewhichbondtobondisgivena(CF),whichmeansaspecificbondisequivalenttoCFbondunderlyinginfuturescontract.Theshortdesignateswhichbondhewilldelivercheapest-to-deliverbond).aspecificBondA:1标准AA3BondpriceisusuallyascleanpriceCleanprice=fullprice-accruedthepricecanwrittenas(S0)RTS00Rf)TIfSisgivencleanprice(quotedprice)f0(S)R)T0Ifthepriceisascleanprice0fthatAIIT(S)R)TAITT0f00fThepriceiswith1S+AIR)TAIFVCQFP00fTCF3ExampleEuro-bundfutureshaveacontractandtheunderlyingconsistslong-termGermandebtinstrumentswithyearsTheytradedontheEurex.Supposethe1underlyingGermanbundquotedatandhasaccruedinterest(one-halfamonthsincelastcoupon).Theeuro-bundfuturescontractmaturesonemonth.contractexpiration,theunderlyingbundhaveaccruedinteresttherenocouponpaymentsdueuntilafterthefuturescontractexpires,andthecurrentone-monthrisk-freerate0.1%.Theconversionfactor01=Inthiscase,wehaveT=CF(T)=B+Y)=0FVCI=AI==AI==andr00T0.1%.Theequilibriumeuro-bundfuturespricebasedonthecarryarbitragemodelwillbeclosestto:A.C.3Example-Solution:Bcorrect.Thecarryarbitragemodelforforwardsandfuturessimplythefuturetheunderlyingwithadjustmentsforuniquecarryfeatures.Withbondfutures,theuniquefeaturesincludetheconversionaccruedinterest,andanycouponpayments.Thus,theequilibriumeuro-bundfuturespricecanbefoundusingthecarryarbitragemodelwhichF(T)=FV–AIT–FVCI0or(T)=[1/CF(T)]{FV+Y)+AI]–AI–FVCI}000TThus,wehaveQF(T)=++––Inequilibrium,theeuro-bundfuturespriceshouldbebasedonthecarryarbitragemodel.=€3Exampleidentifiesanarbitrageopportunityafixed-incomefuturescontractandunderlyingbond.CurrentdataonthefuturescontractandunderlyingbondpresentedExhibit.Thecurrentannualcompoundedrisk-freerate3ExampleBasedarbitrageisclosestB.C.Bno-arbitragepriceisequalfollowing:FT，TTBTYAI0PVCI，T00110.250FT00.25112.08FT0Theadjustedpriceisequalconversionprice:FTTT00FT=0Addingaccruedinexpiration)adjustedpricegivesapriceThisdifferencemeansisoverpriced–=00arbitrageisdifference:=41.FrameworkPrincipleofArbitrage-freePricingEquityandContractsandContracts(FRA)Fixed-IncomeandContractsCurrencyContracts23.T-bond.ContractsCurrencyContractsEquityContracts4aAplainvanillaisinwhichonepartyaandtheotherafloatingPricingaplainvanillameanscalculatingthe()thatthecontractinitiation.Sinceabondhasaequaltoparwilldoistofindabondwithaequaltotheparinitiation.Cthecouponofthen-periodbond,4a1=C×B+C×B+C×B+……+C×B+1×B123nnAndthencangettheCn1CBB12RecallthatBnisthewhichisinnIttothattheanswerCaperiodicandmustittogettheannual4ExampleCalculateswapaplainswap.spotR(90-day)=2.5%;R(180-day)=3%;day)=3.5%;R(360-day)=4%.Answer:Stepthediscountfactors:B1=1/(1+2.5%×0.9938;B2=1/(1+3%×=0B3=1/(1+3.5%×0.9744;B4=1/(1+4%×0Steptheperiodicswaprate,C:C=−B==−0.9615)/(0.9938+0.9852+0.9744+0.9615)4ExampleSteptheannualizedswaprate:swaprate=×=swapbespotrates.Soeveryaswapratealwayscheckwithintherangespotexample,getandthiswithinrangeand4%.trickthattheswaprateusuallyveryclosethelastspotratehere).4aTheofaissimplytheofbuyingabondandissuingafloating-ratebond.CompanyXCompanyYfloatingtheabondwillbetheamountitsperiodicwhenissettothe(floating).Thevaluationformula:V(X)B-BVY)B-BswapfixfltOrVswapPV[FswaprateFswaprate]tTt04ExampleCalculatethetheplainswappay-fixedpreviousexampleafterdays.notionalprincipal$1million.daysspotratesR(60-day)=3%;R(150-day)=3.5%;R(240-day)=4%;day)=4.5%.AnswerCalculatethenewdiscountfactorsdayslater:=1/(1+3%×=0.9950;B2=1/(1+3.5%×=0B3=1/(1+4%×=0.9740;B4=1/(1+4.5%×=0Calculatethethefixed-ratebond:P(fixed)=×(0.9950+0.9856+0.9740+0.9604)×4ExampleStepthethefloating-ratebond:0dayswhenthefirstcomes,bondpricewillfirstat62.5%×Sowegetthefloating-ratebondpriceas:P(floating)=×=Steptheswaptothepay-fixedside:V=−P(fixed)]×notionalprincipal=×=4ExampleQ1:Supposeyoupricingafive-yearLibor-basedinterestrateswap.Theestimatedpresentfactorsasfollows.Calculatethefixedswaprate.0.990099123450.9778760.9651360.9515290.937467CorrectAnswer:511CBB12Becausea5-yearannually-payplainswap,rateequal4ExampleQ2:Supposetwoyearsagoweentereda7-yearreceive-fixedLibor-basedinterestrateswapwithannualresetsdayaccount).Thefixedratetheswapcontractenteredtwoyearsagowas2%.ThepresentfactorssameasaspresentedCalculatetheforthepartytheswapreceivingthefixedrate.CorrectAnswer:theswapperdollarnotionalV1.2968%Thus,thetheswap5acurrencyswapConsideraswapinvolvingtwocurrencies,theUSdollarandtheEuroTheexchangeratenow/$.Aswedoaplainswap,wecangetthefixedratethatwillthefixed$paymentsequalandthefixedratethatwillthefixed€paymentsequaltoForexample,IftheUStermstructureis:R(90-day)=5.2%,R(180-day)=5.4%,day)=5.55%,R(360-day)=5.7%,wecangetthefixedrateIftheEurotermstructureis:R(90-day)=3.45%,R(180-day)=3.58%,R(270-day)=3.7%,R(360-day)=3.75%,wecangetthefixedrateOurcurrencyswapinvolvingdollarsforEuroswouldhaveafixedratedollarsandEuros.ThenotionalprincipalwouldbeandTherefourwaysconstructtheswap:5atothe1:dollar5.56%andreceiveEuro3.68%.2:dollar5.56%andreceiveEurofloating.3:dollarfloatingandreceiveEuro3.68%.4:dollarfloatingandreceivefloating.Inthe4(floatingfloating),isnopricingproblembecauseisnoshouldonlysetthenotionalprincipalto€0.8every1$.5ExampleSomecountrieswillchange,theexchangewillalsobedifferent.cancalculatefixed-andcurrencies,andwillgettheswapjustasyoudoaplainswap.GBP6%CompanyBCompanyAUSD5%thevalueoftheswapinUSDtoCompanyAis:VUSD)B-(SB)0where:S0spotrateinUSDperGBP5ExampleConsideratwo-yearcurrencyswapwithsemiannualpayments.ThedomesticcurrencytheU.S.andtheforeigncurrencytheU.K.pound.Thecurrentexchangerateperpound.L=L=L=L=L₤=₤=L₤=₤=Thecomparableset₤ratesA.Calculatetheannualizedfixedratesfordollarsandpounds:5ExampleAnswerforA:Firstfixedindollarsdollarfor180,360,540,720follows=0₤=11===1801803601+(0.0585)1+(0.0493))360=0₤=11=36060360601+(0.0605)1+(0.050533=0₤=11==540605403601+(0.0624)1+(0.0519)3=0₤=11=0.8826=7207203601+(0.0551)1+(0.0665)360annualizedfixedper$1isannualizedor0.0632for1-0.8826=0.03160.9716+0.9430+0.9144+0.8826Similarlyannualizedfixed£1is:0.0264or0.0528forannualized5ExampleNowforward120newis$1.35pernewU.S.termis:L(60)=0.0613L₤(60)=0.0517L(240)=0.0629L(420)=0.0653L(600)=0.0697L₤(240)=0.0532₤ratesL₤(420)=0.0568L₤(600)=0.0583B.is$1orinBritishCalculatefollowing₤fixed$fixed;₤receive$fixed₤$5ExampleAnswerforB:Thenewdollarandpoundfactorsforanddaysasfollows:1+(0.061311B==₤===60601))1+(0.0517))3603601B=₤===2402401+(0.06291+(0.0532360360B=1₤=1=4204203601+(0.0653))1+(0.0568))360B=₤=11==0.895960060600601+(0.06971+(0.0583335ExampleInterms$payments:Thepresenttheremainingfixedpaymentsplusthenotionalprincipal++++=Thepresentthepaymentsplushypothetical1notionalprincipaldiscountedbackdays+=$[5ExampleInterms£payments:Thepresenttheremainingfixedpaymentsplusthenotionalprincipal++++=Convertthisamounttheequivalentnotionalprincipalandconvertdollarsatthecurrentexchangerate:**=5ExampleThepresentthepaymentsplushypothetical1notionalprincipal+=Convertdollarsatthecurrentexchangerate**=Themarketbasedonnotionalprincipalasfollows:£1£fixedandreceive$fixed==-£andreceive$fixed==-£andreceive$==-£fixedandreceive$==-6ExampleAUScompanyneededborrowAustraliandollarsforoneyearforAustraliansubsidiary.ThecompanydecidedborrowUSdollarsanamountequivalentbyissuingUS-denominatedbonds.Thecompanyenteredintoaone-yearcurrencyswapwithaswapTheswapusesquarterlyresetdaycount)andexchangenotionalamountsatandattheexpiration,theUScompanythenotionalamountAustraliandollarsandreceivesthedealerthenotionalamountUSdollars.ThefixedrateswereforAustraliandollarsandforUSdollars.thenotionalamountUSdollarswaswithaspotexchangerateforUS$1.6ExampleAssumedayshavepassedandweobservethefollowingmarketinformation:A$SpotInterestRates(%)US$SpotInterestRates(%)DaystoMaturityPresent(A$1)Present(US$1)30Sum:Sum:6ExampleThecurrenttheswapdealerA$thecurrencyswapenteredintodaysagowillbeclosestto:A.–C.6ExampleQ1CorrectAnswer:C.FlowsExchanged=A$USFirmQuarterlyFlowsExchangedDealerDealer=USFirmFlowsExchanged1US$87,719,298USFirmDealer6ExampleQ1CorrectAnswer:C.TheUSfirmissuesabondandentersaswapwiththeswapTheexchangerategivenasforTheswapdealerreceivingquarterlyinterestpaymentscurrencyA$.Afterdaysthenewexchangerateperandthetermstructurehaschangedbothmarkets.ThisthetheswapthepartyreceivinginterestpaymentsAustraliandollars,whichtheswap6ExampleThePVthedealerincomingcashflowsA$,effectivelyalongpositionA$bond.m×[)×(++ThePVtheUSDoutflows(effectivelyashortUSDbondThePVthequarterlyinterestpaymentsandterminalpaymentcalculatedusingthenewtermstructureandconvertedinto×(USD)×[()×(0.999584+0.998668+0.998253+0.998336)+=A$dealer-=A$6ExampleThecurrentUSDtheUSfirmthecurrencyswapenteredintodaysagowillbeclosestto:A.––C.6ExampleQ2CorrectAnswer:B.ThetheUSfirm–dealerThisrepresentsthethefirmmakinginterestpaymentsCurrencya(A$)Vfirmdealer=-A$whichconvertedUSD–×=–NotethattheUScompanyissues(short)abondUSDtheirhomemarketandusesaswapeffectivelyconvertbondissue.Understandingtheswapastwobonds,theUSfirmlongaUSDbondandshortabondA$.TheswapoffsetstheUSUSDbondissue(short).TheswapallowstheUSfirmA$interestpaymentstheswaporeffectivelyissueabondA$.6antypesofequity(1)andreceiveequity;(2)floatingandreceiveequity;(3)oneequityreturnandreceiveanotherequity.onlyneedtopricethetypebecausenointheothertwo.thesameformulathetogettheofequityn1CBBB12niscase?canabondofCawiththenotionalamountequaltotheparthebond,becausethebondinceptionisequalto6anequityswapAnequityhas3.92%the$1million.underlyingisindex,currentlytrading1000.after30index1100LIBORratesR(150-day)=3.5%;R(240-day)=4%;R(330-day)=4.5%.Calculateequityfixed-rate7ExampleAnswer:Stepthenewdiscountfactorsdayslater:1=1/(1+3%×=;2=1/(1+3.5%×=3=1/(1+4%×=;0B4=1/(1+4.5%×=0Stepthethefixed-ratebond:P(fixed)=0.98%××=Stepthetheindexinvestment:P(index)=1×=1.1Stepthetothefixed-ratepayer:V=[P(index)−P(fixed)]×notionalprincipal=×=7Aisoptiontointoa.willontheswaption.issimilartoaswaptionmaturesin2andtheholderrighttoa3-theendofthesecondisa2×5swaption.Apayerswaptionisoptiontointoatheincreases,theswaptionup.Soaswaptionaputoptiononacouponbond.alsoaAreceiverswaptionoptiontoathereceiver(theincreases,receivergodown.areceiverswaptionisequivalenttoacalloptiononacouponbond.7aplainistounderlyingiftheislowerthanexpiration,theofaplainvanillaswaptionwillbeExample:Aquarterly-payswaptionwiththeof3.84%andthenotional$1millioncomestoitsLIBORspotR(90-day)=2.5%;R(270-day)=3.5%;ThecurrentonisCalculatetheoftheswaption.7Answer:Step1:Calculatethediscountfactors:B=1/(1+2.5%×90/360)=0.9938;B=1/(1+3%×180/360)=0.985212B=1/(1+3.5%×270/360)=0.9744;B=1/(1+4%×360/360)=0.961534Step2:Calculatethenetcashsavingsateachpaymentdate:(3.92%3.84%)×90/360×million=$200Step3:Calculatethepresentvalueofthesavings:$200×(0.9938+0.9852+0.9744+0.9615)=$783TheownercantheandaOrhecantoreceivequarterlyof$200counterpart.Hecanalsotoreceive$783tothecontract.Iftheis4%inthetheoftheunderlyingis$783andtheswaptionholderwillnottheswaption740Claims71.BinomialModel——TheExpectationsApproachFrameworkTheoneperiodbinomialstockmodelTheperiodbinomialstockmodelbinomialmodel2345.Black-Scholes-MertonModel.OptionandImpliedVolatility.BinomialModel——TheNo-arbitrageApproach.BlackOptionModelOptionOnOption7Afiduciarycallisaportfolioconsistingof:AlongpositioninaEuropeancalloptionwithpriceofXthatmaturitiesinTonastock.Alongpositioninapure-discountrisklessbondthatXinTafiduciarycallisthecostofthecall)plusthecostofthebond(thepresentoftoafiduciarycallwillbeXifthecallisout-of-the-moneyandTifthecallisshowninthefollowing:SSTTCallout-ofor)Callisin-the-money)LongcallLongbond0XXT-XXT7Aprotectiveisaportfolioconsisting:AlongpositioninaEuropeanputoptionwithpriceofXthatmaturitiesinTonastock.Alongpositionintheunderlyingstock.aprotectiveisthecostoftheput)plusthecostofthetoaprotectiveisXiftheputisin-the-moneyandiftheputisout-of-the-money.showninthefollowing:S<X(putisin-the-money)S≥XTT(putisout-oforat-the-money)LongputLongstockX-TT0TXT7syntheticinstruments:XCPSAsyntheticEuropeancalloption:000TR)fsyntheticcall=+−XPCS0AsyntheticEuropeanputoption:00TR)fsynthetic=call+−XPSCAsyntheticpure-discountrisk-lessbond:T000R)fsynthetic=+−callXSCPAsyntheticstockposition:000TR)fsynthetic=call+−7twomighttosyntheticpositionsinthesecurities.priceoptionsbyusingcombinationsoftheotherinstrumentswithknownprices.earnarbitrageprofitsbyexploitingrelativemispricingamongthefoursecurities.Ifput-callparitydoesn’thold,arbitrageprofitis8put-callAswithallarbitragetrades,wantlowsellhigh.”calldoesn’tholdcostafiduciarycalldoesnotequalcostaput),you(golongin)theunderpricedpositionandsell(goshort)theoverpricedposition.Example:Exploitviolationsput-callEuropeancallputwithapricedatandTheunderlyingpricedatandmakesnocashtheoptions.risk-free5%.Calculatethepricecallandillustratehowearnanarbitrage8put-callAnswer:C=P+S−X/(1+RT=−=000fSincethecalloverpricedweshouldthecallforandbuythecallcall,buyforbuytheunderlyingforandshort)awithafaceThetransactionwillgenerateanarbitrage8Abinomialmodelisbasedontheideathat,overtheperiod,somewillchangetooneoftwopossible(binomial).constructabinomialmodel,needtoknowthebeginningassettheofthetwopossiblechanges,andtheprobabilitiesofeachofthesechangesoccurring.offbyhavingonlyonebinomialperiod,whichmeansthattheunderlyingpricemovestotwonewpricesoptionexpiration.let0bethepriceoftheunderlyingstockOneperiodthestockpricecanmoveupto+ordownto−.thenidentifyau,theupmoveonthestockanddthedownmove.Thus,S+uandS−d.furtherassumethatu=1/d.1010+=Su●0S0●S−d●108Risk-neutralprobabilityofupmoveisπu;Risk-neutralprobabilityofdownmoveisπ=1-π:duRfdud1uwithacallthegoesupto+,thecalloptionbeC+.downtoS−,theoptionbeC−.knowthat111theofacalloptionbeonThusget:C+=Max(0,S+;C−=Max(0,S−11111option:cCC11udR)TfC+1+1-C-hhedgeratio=1(sharesperoption)S-18ExampleCalculatethetodaya1-yearcalloptiononthestockwiththepriceThepricethestockandthesizeanup-moveTherisk-freerate7%.Answer:Calculatetheparameters:u=1.25;d=1/u=0.8;S×S×udC+=Max=5;C−=Max=0Calculaterisk-neutralπandπ=1−π:uduπu==π=1−π=0.4duCalculatecalloptionprice:C08ExampleDrawtheone-periodbinomialtree:+=SC+=5(C=?)S=20.40−=SdC−=0todayyear18ExamplePricingaputoptionsimilarthatacall.TheonlydifferencethatP+=MaxX−S)andP−=MaxX−SExample:UsetheinformationthepreviousexamplecalculatethetodayaputonthesamestockwiththepriceAnswer:P+=Max=P−=Max=4P=×0+×==8S++=uu++=Max(0,++S+=u+SS+−=−+=S=S=Suddu(C=?)+−=C−+=Max(0,+−−=dC−S−−=dd−−=Max(0,−−218ExampleCalculateacalloptiononstockwithpriceThethestockandsizeanup-moveTherisk-freerate7%.8ExampleAnswer:Calculatetheparameters:U=d=1/u=S=×=S=×=udS=××=S=Sdd=××=C=Max−=C=Max−=2C=Max−=0Calculaterisk-neutralπand−π:πu=−−=πd=−π=9AaStep3:thetwo-periodbinomialtree:++=31.25C++=13.25S+=25C+S=20(C=?)+−=20C+−−+=2S−=16C−−−=12.8C−−=0129AaStep4:Calculatethecalloptionat1:C+=(13.25×0.6+2×0.4)/1.07=8.1776C−=(2×0.6+0×0.4)/1.07=1.1215Step5:CalculatethecalloptionC=(8.1776×0.6+1.1215×0.4)/1.07=5.00489ExampleConsideratwo-periodbinomialmodelwhichastocktradesatapricestockgouppercentor7percenteachperiod.5percent.priceaputoptionexpiringtwoperiodswithexerciseprice19ExampleAnswer:Calculatetheparameters:U=d=Su=×=Sd=×=S=××=S=××=Sdd=××=P=Max=0P=Max-=0P=Max=Calculaterisk-neutralπand−π:πu=−−0.83)=πd=−π=9ExampleDrawthetwo-periodbinomialtree:S=SP=0+=S=78P+S=65(P=?)S=P=0S−=SdP−S=Sdd=year2todayyear19ExampleCalculatetheputoptionatyearP+=×+×=0P−=×+×Calculatetheputoptiontoday:P=×+×9(AmericanS++=SC++=Max(0,S++−X)+=SuC+SS+−=S−+=S=S=S(C=?)+−=C−+=Max(0,S+−−X)S−=SdC−−−=SC−−=Max(0,S−−−X)129Exampleobservea€50priceforastock.calloptiontwoperiodicallyrisk-freeis5%,priceis€50,u=1.356,d=0.744.calloptionis1.moveonrisk-neutralisA.30%.B.40%.C.50%.23.calloptionisto:A.€9.53.B.€9.71.C.€9.87..optionisto:A.€5.06.B.€5.33.C.€5.94.9Example-Solutionto1:CisBasedonRNweπ=––=+0.05)–0.744]/(1.356–0.744)=50%Solutionto2:Biscalloptioncalculationsfollows:=S–=Max[0,1.356(50)–50]=41.9368+=–=–=–50]=0.44320––=S–=(50)–50]=Withthisinformation,wecalloptionvalue:c==PV[π+––+––]==+41.9368+–0.5)0.44320+–0.0]9.71Solutionto3:Aisoptionsimplyapplying–callorp=c+–S=9.71++50–50=.06.priceis€5.06.59ThebinomialasetofpossiblepathsthatusetooptionsonbondsorBinomialeiuuiui0i)udduididd121don’tneedtohowtoconstr