CFA特許金融分析師-CFA二級-Derivatives

CFA特許金融分析師-CFA二級-Derivatives共享題干題SonalJohnsonisariskmanagerforabank.Shemanagesthebank'sris（江南博哥）ksusingacombinationofswapsandforwardrateagreements(FRAs).Johnsonpricesathree-yearLibor-basedinterestrateswapwithannualresetsusingthepresentvaluefactorspresentedinExhibit1.JohnsonalsousesthepresentvaluefactorsinExhibit1tovalueaninterestrateswapthatthebankenteredintooneyearagoasthepay-fixed(receive-floating)party.SelecteddatafortheswaparepresentedinExhibit2.Johnsonnotesthatthecurrentequilibriumtwo-yearfixedswaprateis1.12%.Oneofthebank'sinvestmentsisexposedtomovementsintheJapaneseyen,andJohnsondesirestohedgethecurrencyexposure.Shepricesaone-yearfixed-for-fixedcurrencyswapinvolvingyenandUSdollars,withaquarterlyreset.JohnsonusestheinterestratedatapresentedinExhibit3topricethecurrencyswap.Johnsonnextreviewsanequityswapwithanannualresetthatthebankenteredintosixmonthsagoasthereceive-fixed,pay-equityparty.Selecteddataregardingtheequityswap,whichislinkedtoanequityindex,arepresentedinExhibit4.Atthetimeofinitiation,theunderlyingequityindexwastradingat100.00.Theequityindexiscurrentlytradingat103.00,andrelevantUSspotrates,alongwiththeirassociatedpresentvaluefactors,arepresentedinExhibit5.Johnsonreviewsa6×9FRAthatthebankenteredinto90daysagoasthepay-fixed/receive-floatingparty.SelecteddatafortheFRAarepresentedinExhibit6,andcurrentLibordataarepresentedinExhibit7.Basedonherinterestrateforecast,Johnsonalsoconsiderswhetherthebankshouldenterintonewpositionsin1×4and2×5FRAs.Threemonthslater,the6×9FRAinExhibit6reachesexpiration,atwhichtimethethree-monthUSdollarLiboris1.10%andthesix-monthUSdollarLiboris1.20%.JohnsondeterminesthattheappropriatediscountratefortheFRAsettlementcashflowsis1.10%.[單選題]1.BasedonExhibit1,Johnsonshouldpricethethree-yearLibor-basedinterestrateswapatafixedrateclosestto:A.0.34%.B.1.16%.C.1.19%.正確答案：C參考解析：TheswappricingequationisThatis,thefixedswaprateisequalto1minusthefinalpresentvaluefactor(inthiscase,Year3)dividedbythesumofthepresentvalues(inthiscase,thesumofYears1,2,and3).ThesumofpresentvaluesforYears1,2,and3iscalculatedas[單選題]2.Fromthebank'sperspective,usingdatafromExhibit1,thecurrentvalueoftheswapdescribedinExhibit2isclosestto:A.–$2,951,963.B.–$1,849,897.C.–$1,943,000.正確答案：B參考解析：Thevalueofaswapfromtheperspectiveofthereceive-fixedpartyiscalculatedasTheswaphastwoyearsremaininguntilexpiration.Thesumofthepresentval?uesforYears1and2isGiventhecurrentequilibriumtwo-yearswaprateof1.00%andthefixedswaprateatinitiationof3.00%,theswapvalueperdollarnotionaliscalculatedasV=(0.03–0.0112)×1.967975=0.0336998Thecurrentvalueoftheswap,fromtheperspectiveofthereceive-fixedparty,is$50,000,000×0.035998=$1,849,897.Fromtheperspectiveofthebank,asthereceive-floatingparty,thevalueoftheswapis–$1,849,897.[單選題]3.BasedonExhibit3,JohnsonshoulddeterminethattheannualizedequilibriumfixedswaprateforJapaneseyenisclosestto:A.0.0624%.B.0.1375%.C.0.2496%.正確答案：C參考解析：TheequilibriumswapfixedrateforyeniscalculatedasTheyenpresentvaluefactorsarecalculatedas90-dayPVfactor=1/[10.0005(90/360)]=0.999875.180-dayPVfactor=1/[10.0010(180/360)]=0.999500.270-dayPVfactor=1/[10.0015(270/360)]=0.998876.360-dayPVfactor=1/[10.0025(360/360)]=0.997506.Sumofpresentvaluefactors=3.995757.Therefore,theyenperiodicrateiscalculatedasTheannualizedrateis(360/90)timestheperiodicrateof0.0624%,or0.2496%.[單選題]5.BasedonExhibit5,thecurrentvalueoftheequityswapdescribedinExhibit4wouldbezeroiftheequityindexwascurrentlytradingtheclosestto:A.97.30.B.99.09.C.100.00.正確答案：B參考解析：Theequityindexlevelatwhichtheswap'sfairvaluewouldbezerocanbecalculatedbysettingtheswapvaluationformulaequaltozeroandsolvingforSt:Thevalueofthefixedlegoftheswaphasapresentvalueof$19,818,677,or99.0934%ofparvalue:TreatingtheswapnotionalvalueasparvalueandsubstitutingthepresentvalueofthefixedlegandS0intotheequationyields[單選題]6.Fromthebank'sperspective,basedonExhibits6and7,thevalueofthe6×9FRA90daysafterinceptionisclosestto:A.$14,817.B.$19,647.C.$29,635.正確答案：A參考解析：Thecurrentvalueofthe6×9FRAiscalculatedasThe6×9FRAexpiressixmonthsafterinitiation.ThebankenteredintotheFRA90daysago;thus,theFRAwillexpirein90days.TovaluetheFRA,thefirststepistocomputethenewFRArate,whichistherateonDay90ofanFRAthatexpiresin90daysinwhichtheunderlyingisthe90-dayLibor,orFRA(90,90,90):Exhibit7indicatesthatL90(180)=0.95%andL90(90)=0.90%,soFRA(90,90,90)={[10.0095(180/360)]/[10.0090(90/360)]-1}/(90/360)FRA(90,90,90)=[(1.00475/1.00225)-1](4)=0.009978,or0.9978%Therefore,giventheFRArateatinitiationof0.70%andnotionalprincipalof$20millionfromExhibit1,thecurrentvalueoftheforwardcontractiscalculatedasVg(0,h,m)=V90(0,180,90)V90(0,180,90)=$20,000,000[(0.009978-0.0070)(90/360)]/[10.0095(180/360)].V90(0,180,90)=$14,887.75/1.00475=$14,817.37.[單選題]7.BasedonExhibit7,theno-arbitragefixedrateonanew1×4FRAisclosestto:A.0.65%.B.0.73%.C.0.98%.正確答案：C參考解析：Theno-arbitragefixedrateonthe1×4FRAiscalculatedasFora1×4FRA,thetworatesneededtocomputetheno-arbitrageFRAfixedrateareL(30)=0.75%andL(120)=0.92%.Therefore,theno-arbitragefixedrateonthe1×4FRArateiscalculatedasFRA(0,30,90)={[10.0092(120/360)]/[10.0075(30/360)]-1}/(90/360).FRA(0,30,90)=[(1.003066/1.000625)-1]4=0.009761,or0.98%rounded.[單選題]8.BasedonExhibit7,thefixedrateonanew2×5FRAisclosestto:A.0.61%.B.1.02%.C.1.71%.正確答案：B參考解析：Thefixedrateonthe2×5FRAiscalculatedasFora2×5FRA,thetworatesneededtocomputetheno-arbitrageFRAfixedrateareL(60)=0.82%andL(150)=0.94%.Therefore,theno-arbitragefixedrateonthe2×5FRArateiscalculatedasFRA(0,60,90)={[10.0094(150/360)]/[10.0082(60/360)]-1}/(90/360)FRA(0,60,90)=[(1.003917/1.001367)-1]4=0.010186,or1.02%rounded[單選題]9.BasedonExhibit6andthethree-monthUSdollarLiboratexpiration,thepaymentamountthatthebankwillreceivetosettlethe6×9FRAisclosestto:A.$19,945.B.$24,925.C.$39,781.正確答案：A參考解析：Givenathree-monthUSdollarLiborof1.10%atexpiration,thesettlementamountforthebankasthereceive-floatingpartyiscalculatedasTherefore,thebankwillreceive$19,945(rounded)asthereceive-floatingparty.TridentAdvisoryGroupmanagesassetsforhigh-net-worthindividualsandfamilytrusts.AliceLee,chiefinvestmentofficer,ismeetingwithaclient,NoahSolomon,todiscussriskmanagementstrategiesforhisportfolio.SolomonisconcernedaboutrecentvolatilityandhasaskedLeetoexplainoptionsvaluationandtheuseofoptionsinriskmanagement.OptionsonStockLeebegins:“WeusetheBlack-Scholes-Merton(BSM)modelforoptionvaluation.TofullyunderstandtheBSMmodelvaluation,oneneedstounderstandtheassumptionsofthemodel.Theseassumptionsincludenormallydistributedstockreturns,constantvolatilityofreturnontheunderlying,constantinterestrates,andcontinuousprices”LeeusestheBSMmodeltopriceTCB,whichisoneofSolomon'sholdings.Exhibit1providesthecurrentstockprice(S),exerciseprice(X),risk-freeinterestrate(r),volatility(σ),andtimetoexpiration(T)inyearsaswellasselectedoutputsfromtheBSMmodel.TCBdoesnotpayadividend.OptionsonFuturesTheBlackmodelvaluationandselectedoutputsforoptionsonanotherofSolomon'sholdings,theGPX500Index(GPX),areshowninExhibit2.ThespotindexlevelfortheGPXis187.95,andtheindexisassumedtopayacontinuousdividendatarateof2.2%(5)overthelifeoftheoptionsbeingvalued,whichexpirein0.36years.AfuturescontractontheGPXalsoexpiringin0.36yearsiscurrentlypricedat186.73.AfterreviewingExhibit2,SolomonasksLeewhichoptionGreekletterbestdescribesthechangesinanoption'svalueastimetoexpirationdeclines.SolomonobservesthatthemarketpriceoftheputoptioninExhibit2is$7.20.LeerespondsthatsheusedthehistoricalvolatilityoftheGPXof24%asaninputtotheBSMmodel,andsheexplainstheimplicationsfortheimpliedvolatilityfortheGPX.OptionsonInterestRatesSolomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]10.BasedonExhibit1andtheBSMvaluationapproach,theinitialportfoliorequiredtoreplicatethelongcalloptionpayoffis:A.Long0.3100sharesofTCBstockandshort0.5596sharesofazero-couponbond.B.Long0.6217sharesofTCBstockandshort0.1500sharesofazero-couponbond.C.Long0.6217sharesofTCBstockandshort0.5596sharesofazero-couponbond.正確答案：C參考解析：Theno-arbitrageapproachtocreatingacalloptioninvolvesbuyingDelta=N(d1)=0.6217sharesoftheunderlyingstockandfinancingwith–N(d2)=-0.5596sharesofarisk-freebondpricedatexp(-rt)(X)=exp(-0.0022x0.25)(55)=$54.97perbond.NotethatthevalueofthisreplicatingportfolioisnsS+nBB=0.6217(57.03)-0.5596(54.97)=$4.6943(thevalueofthecalloptionwithslightroundingerror).Solomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]11.TodeterminethelongputoptionvalueonTCBstockinExhibit1,thecorrectBSMvaluationapproachistocompute:A.0.4404timesthepresentvalueoftheexercisepriceminus0.6217timesthepriceofTCBstock.B.0.4404timesthepresentvalueoftheexercisepriceminus0.3783timesthepriceofTCBstock.C.0.5596timesthepresentvalueoftheexercisepriceminus0.6217timesthepriceofTCBstock.正確答案：B參考解析：TheformulafortheBSMpriceofaputoptionisp=e-rtXN(-d2)-SN(-d1).N(-d1)=1-N(d1)=1-0.6217=0.3783,andN(-d2)=1-N(d2)=1-0.5596=0.4404.NotethattheBSMmodelcanberepresentedasaportfolioofthestock(nsS)andzero-couponbonds(NBB).Foraput,thenumberofsharesisns=-N(-d1)＜0andthenumberofbondsisnB=-N(d2)>0.ThevalueofthereplicatingportfolioisnSS+nBB=-0.3783(57.03)+0.4404(54.97)=$2.6343(thevalueoftheputoptionwithslightroundingerror).Bisarisk-freebondpricedatexp(-rt)(X)=exp(-0.0022x0.25)(55)=$54.97.Solomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]12.Whatarethecorrectspotvalue(S)andtherisk-freerate(r)thatLeeshoulduseasinputsfortheBlackmodel?A.186.73and0.39%,respectivelyB.186.73and2.20%,respectivelyC.187.95and2.20%,respectively正確答案：A參考解析：Black'smodeltovalueacalloptiononafuturescontractisc=e-rT[F0(T)N(d1)-XN(d2)].TheunderlyingF0isthefuturesprice(186.73).Thecorrectdiscountrateistherisk-freerate,r=0.39%.Solomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]13.WhichofthefollowingisthecorrectanswertoSolomon'squestionregardingtheoptionGreekletter?A.VegaB.ThetaC.Gamma正確答案：B參考解析：Leeispointingouttheoptionprice'ssensitivitytosmallchangesintime.IntheBSMapproach,optionpricesensitivitytochangesintimeisgivenbytheoptionGreektheta.Solomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]14.BasedonSolomon'sobservationaboutthemodelpriceandmarketpricefortheputoptioninExhibit2,theimpliedvolatilityfortheGPXismostlikely:A.Lessthanthehistoricalvolatility.B.Equaltothehistoricalvolatility.C.Greaterthanthehistoricalvolatility.正確答案：A參考解析：Theputispricedat$7.4890bytheBSMmodelwhenusingthehistoricalvolatilityinputof24%.Themarketpriceis$7.20.TheBSMmodeloverpricingsuggeststheimpliedvolatilityoftheputmustbelowerthan24%.Solomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]15.ThevaluationinputsusedbyLeetopriceacallreflectingSolomon'sinterestrateviewsshouldincludeanunderlyingFRArateof:A.0.60%withsixmonthstoexpiration.B.0.75%withninemonthstoexpiration.C.0.75%withsixmonthstoexpiration.正確答案：C參考解析：Solomon'sforecastisforthethree-monthLibortoexceed0.85%insixmonths.Thecorrectoptionvaluationinputsusethesix-monthFRArateastheunderlying,whichcurrentlyhasarateof0.75%.Solomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]16.ThestrategysuggestedbyLeeforhedgingsmallmovesinSolomon'sETFpositionwouldmostlikelyinvolve:A.Sellingputoptions.B.Sellingcalloptions.C.Buyingcalloptions.正確答案：B參考解析：BecausesellingcalloptionscreatesashortpositionintheETFthatwouldhedgehiscurrentlongpositionintheETF.Exhibit2couldalsobeusedtoanswerthequestion.Solomonowns10,000sharesoftheGPX,eachwithadeltaof+1;bydefinition,hisportfoliodeltais+10,000.Adeltahedgecouldbeimplementedbysellingenoughcallstomaketheportfoliodeltaneutral:Solomonforecaststhethree-monthLiborwillexceed0.85%insixmonthsandisconsideringusingoptionstoreducetheriskofrisingrates.HeasksLeetovalueaninterestratecallwithastrikepriceof0.85%.Thecurrentthree-monthLiboris0.60%,andanFRAforathree-monthLiborloanbeginninginsixmonthsiscurrently0.75%.HedgingStrategyfortheEquityIndexSolomon'sportfoliocurrentlyholds10,000sharesofanexchange-tradedfund(ETF)thattrackstheGPX.Heisworriedtheindexwilldecline.HeremarkstoLee,“YouhavetoldmehowtheBSMmodelcanprovideusefulinformationforreducingtheriskofmyGPXposition”LeesuggestsadeltahedgeasastrategytoprotectagainstsmallmovesintheGPXIndex.LeealsoindicatesthatalongpositioninputscouldbeusedtohedgelargermovesintheGPX.Shenotesthatalthoughhedgingwitheitherputsorcallscanresultinadelta-neutralposition,theywouldneedtoconsidertheresultinggamma.[單選題]17.Lee'sput-basedhedgestrategyforSolomon'sETFpositionwouldmostlikelyresultinaportfoliogammathatis:A.Negative.B.Neutral.C.Positive.正確答案：C參考解析：Becausethegammaofthestockpositionis0andtheputgammaisalwaysnon-negative,addingalongpositioninputoptionswouldmostlikelyresultinapositiveportfoliogamma.Gammaisthechangeindeltafromasmallchangeinthestock'svalue.Astockpositionalwayshasadeltaof+1.Becausethedeltadoesnotchange,gammaequals0.Thegammaofacallequalsthegammaofasimilarput,whichcanbeprovenusingput–callparity.BrunoSousahasbeenhiredrecentlytoworkwithsenioranalystCamilaRocha.Rochagiveshimthreeoptionvaluationtasks.AlphaCompanySousa'sfirsttaskistoillustratehowtovalueacalloptiononAlphaCompanywithaone-periodbinomialoptionpricingmodel.Itisanon-dividend-payingstock,andtheinputsareasfollows.●Thecurrentstockpriceis50,andthecalloptionexercisepriceis50.●Inoneperiod,thestockpricewilleitherriseto56ordeclineto46.●Therisk-freerateofreturnis5%perperiod.Basedonthemodel,RochaasksSousatoestimatethehedgeratio,therisk-neutralprobabilityofanupmove,andthepriceofthecalloption.Intheillustration,Sousaisalsoaskedtodescriberelatedarbitragepositionstouseifthecalloptionisoverpricedrelativetothemodel.BetaCompanyNext,Sousausesthetwo-periodbinomialmodeltoestimatethevalueofaEuropean-stylecalloptiononBetaCompany'scommonshares.Theinputsareasfollows.●Thecurrentstockpriceis38,andthecalloptionexercisepriceis40.●Theupfactor(u)is1.300,andthedownfactor(d)is0.800.●Therisk-freerateofreturnis3%perperiod.Sousathenanalyzesaputoptiononthesamestock.Alloftheinputs,includingtheexerciseprice,arethesameasforthecalloption.HeestimatesthatthevalueofaEuropean-styleputoptionis4.53.Exhibit1summarizeshisanalysis.SousanextmustdeterminewhetheranAmerican-styleputoptionwouldhavethesamevalue.Exhibit1.Two-periodBinomialEuropean-StylePutOptiononBetaCompanySousamakestwostatementswithregardtothevaluationofaEuropean-styleoptionundertheexpectationsapproach.Statement1:Thecalculationinvolvesdiscountingattherisk-freerate.Statement2:Thecalculationusesrisk-neutralprobabilitiesinsteadoftrueprobabilities.RochaasksSousawhetheritiseverprofitabletoexerciseAmericanoptionspriortomaturity.Sousaanswers,“Icanthinkoftwopossiblecases.ThefirstcaseistheearlyexerciseofanAmericancalloptiononadividend-payingstock.ThesecondcaseistheearlyexerciseofanAmericanputoption.”InterestRateOptionThefinaloptionvaluationtaskinvolvesaninterestrateoption.Sousamustvalueatwo-year,European-stylecalloptiononaone-yearspotrate.Thenotionalvalueoftheoptionis1million,andtheexerciserateis2.75%.Therisk-neutralprobabilityofanupmoveis0.50.Thecurrentandexpectedone-yearinterestratesareshowninExhibit2,alongwiththevaluesofaone-yearzero-couponbondof1notionalvalueforeachinterestrate.Exhibit2.Two-YearInterestRateLatticeforanInterestRateOptionRochaasksSousawhythevalueofasimilarin-the-moneyinterestratecalloptiondecreasesiftheexercisepriceishigher.Sousaprovidestworeasons.Reason1:Theexercisevalueofthecalloptionislower.Reason2:Therisk-neutralprobabilitiesarechanged.[單選題]18.TheoptimalhedgeratiofortheAlphaCompanycalloptionusingtheone-periodbinomialmodelisclosestto:A.0.60.B.0.67.C.1.67.正確答案：A參考解析：Thehedgeratiorequirestheunderlyingstockandcalloptionvaluesfortheupmoveanddownmove.S+=56,andS-=46.c+=Max(0,S+-X)=Max(0,56-50)=6,andc-=Max(0,S--X)=Max(0,46-50)=0.Thehedgeratiois●Thecurrentstockpriceis38,andthecalloptionexercisepriceis40.●Theupfactor(u)is1.300,andthedownfactor(d)is0.800.●Therisk-freerateofreturnis3%perperiod.Sousathenanalyzesaputoptiononthesamestock.Alloftheinputs,includingtheexerciseprice,arethesameasforthecalloption.HeestimatesthatthevalueofaEuropean-styleputoptionis4.53.Exhibit1summarizeshisanalysis.SousanextmustdeterminewhetheranAmerican-styleputoptionwouldhavethesamevalue.Exhibit1.Two-periodBinomialEuropean-StylePutOptiononBetaCompanySousamakestwostatementswithregardtothevaluationofaEuropean-styleoptionundertheexpectationsapproach.Statement1:Thecalculationinvolvesdiscountingattherisk-freerate.Statement2:Thecalculationusesrisk-neutralprobabilitiesinsteadoftrueprobabilities.RochaasksSousawhetheritiseverprofitabletoexerciseAmericanoptionspriortomaturity.Sousaanswers,“Icanthinkoftwopossiblecases.ThefirstcaseistheearlyexerciseofanAmericancalloptiononadividend-payingstock.ThesecondcaseistheearlyexerciseofanAmericanputoption.”InterestRateOptionThefinaloptionvaluationtaskinvolvesaninterestrateoption.Sousamustvalueatwo-year,European-stylecalloptiononaone-yearspotrate.Thenotionalvalueoftheoptionis1million,andtheexerciserateis2.75%.Therisk-neutralprobabilityofanupmoveis0.50.Thecurrentandexpectedone-yearinterestratesareshowninExhibit2,alongwiththevaluesofaone-yearzero-couponbondof1notionalvalueforeachinterestrate.Exhibit2.Two-YearInterestRateLatticeforanInterestRateOptionRochaasksSousawhythevalueofasimilarin-the-moneyinterestratecalloptiondecreasesiftheexercisepriceishigher.Sousaprovidestworeasons.Reason1:Theexercisevalueofthecalloptionislower.Reason2:Therisk-neutralprobabilitiesarechanged.[單選題]19.Therisk-neuvtralprobabilityoftheupmovefortheAlphaCompanystockisclosestto:A.0.06.B.0.40.C.0.65.正確答案：C參考解析：Forthisapproach,therisk-freerateisr=0.05,theupfactorisu=S+/S=56/50=1.12,andthedownfactorisd=S-/S=46/50=0.92.Therisk-neutral