2025年CFA二級(jí)真題衍生品考試時(shí)間：______分鐘總分：______分姓名：______試卷內(nèi)容：1.Aportfoliomanagerusesalongputoptiontohedgethedownsideriskofastockportfolio.Thestockpriceiscurrently$150pershare,andtheputoptionhasastrikepriceof$140andatimetoexpirationof90days.Therisk-freerateis2%annually,andtheputoptionpremiumis$5pershare.Assumingtheportfolioiswell-hedgedbasedonthedeltaoftheputoption,whatisthemaximumpotentiallosstheportfoliomanagerexpectstoincuroverthenext90days,ignoringthecostoftheputoption?2.TheBlack-Scholes-MertonmodelisusedtovalueaEuropeancalloptiononanon-dividendpayingstock.Thestockpriceis$80,thestrikepriceis$75,therisk-freerateis5%annually,andthevolatilityis30%annually.Theoptionhas6monthstoexpiration.CalculatethetheoreticalvalueofthecalloptionusingtheBlack-Scholes-Mertonformula.3.Aninvestorbuysacalloptionwithastrikepriceof$50andaputoptionwithastrikepriceof$50,bothexpiringin6months.Thecalloptionpremiumis$3,andtheputoptionpremiumis$2.Thecurrentstockpriceis$48.Whatisthemaximumlossandthebreakevenpointforthisstrategy?4.Acompanyisenteringintoa1-yearinterestrateswapwithabank.Thecompanypaysafixedinterestrateof4%annuallyonanotionalamountof$10million,andreceivesafloatinginterestratebasedonSOFR(SecuredOvernightFinancingRate)plus50basispoints.ThecurrentSOFRis2.5%.Assumingnopaymenthasbeenmadeyetandtheswapisfairlyvalued,whatistheestimatedcashflowthatthecompanywillpaytothebankattheendofthefirstyear?5.Atraderholdsalongpositionin100contractsofastockindexfuturescontractwithasettlementpriceof$3,500percontract.Theinitialmarginrequirementis$20,000percontract,andthemaintenancemarginrequirementis$15,000percontract.Themarginaccountcurrentlyhas$25,000.Ifthesettlementpricedropsto$3,300percontractthenextday,willamargincalloccur?Ifso,howmuchadditionalmarginmustthetraderdeposit?6.ConsideraEuropeanputoptiononadividend-payingstock.Thestockpriceis$110,thestrikepriceis$120,therisk-freerateis3%annually,thedividendyieldis1%annually,thevolatilityis40%annually,andtheoptionhas1yeartoexpiration.UsingtheBlack-Scholes-Mertonmodel,calculatetheimpliedvolatilityoftheoptionifitsmarketpriceis$12.7.Aninvestorconstructsalongstraddlestrategyonastockcurrentlytradingat$200.Bothoptionshaveastrikepriceof$200andexpirein3months.Iftheinvestorpaysatotalpremiumof$20forbothoptions(i.e.,$10peroption),whatisthebreakevenpointforthisstrategy?8.Acompanyneedstohedgetheforeignexchangeriskofpaying$1millionin6months,withtheforeigncurrencybeingtheEuro.Thecurrentspotexchangerateis1.10USD/EUR.Thecompanyentersintoa6-monthforwardcontract.Ifthe6-monthforwardpremiumforEUR/USDis50basispoints,whatistheforwardexchangeratethecompanywilllockin?WhatistheexpectedcostinUSDatmaturityiftheforwardcontractisheldtomaturity?9.CalculatetheGammaofaEuropeancalloptionusingtheBlack-Scholes-Mertonmodelforthefollowingparameters:Stockprice=$100,Strikeprice=$95,Risk-freerate=5%annually,Volatility=35%annually,Timetoexpiration=6months.Assumetheoptiondeltais0.5.10.Aninvestmentfirmmanagesaportfoliowithabetaof1.2.Thefirmwantstousefuturescontractsonastockindextoreducetheportfolio'sbetato0.9.Thecurrentindexlevelis1,500,andthe1-yearfuturespriceis1,550.Theinitialmarginrequirementforfuturescontractsis10%ofthefuturesprice.Ifthefirmshorts50contracts,whatwillbetheapproximateinitialmarginrequiredforthisposition?11.Atradersellsacalloptionwithastrikepriceof$60for$4pershare.Thestockpriceiscurrently$58.Thetraderdecidestobuyacalloptionwithastrikepriceof$65for$2persharetoestablisharatiocallspread.Whatisthenetpremiumreceived(orpaid)fromestablishingthisspread?Whatisthebreakevenpointforthespread?12.A5-yearswaphasanotionalprincipalof$50million.Thefixedinterestratepaidbythefloating-ratepayeris4%annually.Thefloatingratesforthenext5yearsareprojectedtobe:Year1:3.0%,Year2:3.5%,Year3:4.0%,Year4:4.5%,Year5:5.0%.Whatistheestimatedtotalcashflowpaidbythefixed-ratepayertothefloating-ratepayeroverthefirst3yearsoftheswap,assumingpaymentsaremadeannually?13.Explaintheconceptof'theta'foracalloption.Howdoesthethetachangeastheoptionapproachesexpiration?Why?14.Acompanyisevaluatingwhethertouseacurrencyswaporaseriesofforwardcontractstomanagetheexchangerateriskassociatedwithfuturecashflowsinaforeigncurrency.Whatarethekeydifferencesbetweenusingacurrencyswapversusaseriesofforwardcontractsforthispurpose?15.Calculatethe1-day99%VaRforaportfolioofoptionspositionsusingtheparametricmethod.Theportfolioconsistsof1,000calloptionswithanaveragedeltaof0.3,anaveragegammaof0.05,avolatilityof20%,andatimetoexpirationof30days.Assumetherisk-freerateis1%annually.Theportfolioiscurrentlyvaluedat$10million.試卷答案：1.$150-140-(5-(150-140)*e^{-0.02*90/365}*N(d_1)+140*e^{-0.02*90/365}*N(d_2))$Whered1,d2,N(d1),N(d2)needtobecalculatedbasedonBSmodelinputs.However,sincetheportfolioisperfectlyhedgedbydelta,themaximumlossistheoreticallylimitedtothebreak-evenpointbelowthestrike,minusthepremiumpaid.Break-evenprice=Strike-Premium=$140-$5=$135.Lossbelow$135=$135-$140+$5=$0.Thisimpliesthedeltahedgeisperfect,andthepremiumcoverstheexpectedloss.Therefore,themaximumlossexpected,consideringthehedgeandpremium,is$0.*(Note:Thisinterpretationassumesthehedgeperfectlyoffsetstheriskuptothebreak-evenpoint,andthepremiumpaidfullyinsuresagainstlossesbelowthatpoint.Amorecomplexscenariomightinvolvegammariskifthestockpricemovessignificantly,butthepromptsimplifiesit)*.Answer:$02.d1=[ln(S/K)+(r+σ2/2)T]/(σ√T)d1=[ln(80/75)+(0.05+0.302/2)*0.5]/(0.30*√0.5)d1=[ln(1.06667)+(0.05+0.045)*0.5]/(0.30*0.7071)d1=[0.0645+0.0475*0.5]/0.2121d1=[0.0645+0.02375]/0.2121d1=0.08825/0.2121≈0.4165d2=d1-σ√T=0.4165-0.30*0.7071≈0.4165-0.2121≈0.2044N(d1)≈0.6602(fromstandardnormaltable)N(d2)≈0.5808(fromstandardnormaltable)CallValue=S*N(d1)-K*e^(-rT)*N(d2)CallValue=80*0.6602-75*e^(-0.05*0.5)*0.5808CallValue=52.816-75*0.9753*0.5808CallValue=52.816-43.946*0.5808CallValue=52.816-25.518≈$27.298Answer:$27.303.ThisisaBullPutSpread.MaxLoss=(LowerStrike-StockPrice)-(PutPremiumPaid-CallPremiumReceived)MaxLoss=($50-$48)-($2-$3)=$2-(-$1)=$3BreakevenPoint(Lower)=LowerStrike-(PutPremiumPaid-CallPremiumReceived)BreakevenPoint(Lower)=$50-($2-$3)=$50+$1=$51BreakevenPoint(Higher)=UpperStrike+(PutPremiumPaid-CallPremiumReceived)BreakevenPoint(Higher)=$50+($2-$3)=$50-$1=$49*(Note:Thequestionasksforthemaximumlossandthebreakevenpoint.Themaximumlossis$3.Thestrategyhastwobreakevenpoints:$49(wherethenetcreditisrecovered)and$51(wherethenetdebitisrecovered).)*Answer:MaximumLoss=$3;BreakevenPoints=$49and$514.FixedPayment=Notional*FixedRate*(DaysinPeriod/365)=$10,000,000*4%*(365/365)=$400,000FloatingPaymentReceived=Notional*(SOFR+Spread)*(DaysinPeriod/365)=$10,000,000*(2.5%+0.50%)*(365/365)=$10,000,000*3.0%=$300,000NetCashFlowPaid=FixedPayment-FloatingPaymentReceived=$400,000-$300,000=$100,000*(Note:Thequestionasksforthecashflowpaid*bythecompany*.Thecompanypaysthefixedrateandreceivesthefloatingrate.Sincetheswapisfairlyvaluedatinitiation,thenetpaymentrepresentsthecompany'snetcostforthefirstperiod.)*Answer:$100,0005.InitialPositionValue=100*$3,500=$350,000PortfolioValueChange=100*($3,300-$3,500)=100*(-$200)=-$20,000NewPortfolioValue=$350,000-$20,000=$330,000EquityinAccount=InitialMargin+NewPortfolioValue-TotalMarginCalls=$25,000+$330,000-0(assumingnopriormargincalls)=$355,000MaintenanceMarginRequirement=100*$15,000=$150,000MarginCalloccursifEquity<MaintenanceMargin.$355,000>$150,000,sonomargincalloccurs.Answer:Nomargincall;Additionalmarginneeded=$06.UsingtheBlack-Scholes-Mertonformulaforaputoption:PutPrice=K*e^(-qT)*N(-d2)-S*e^(-qT)*N(-d1)Whereq=dividendyield,d1=[ln(S/K)+(r-q+σ2/2)T]/(σ√T),d2=d1-σ√TGiven:S=110,K=120,r=0.03,q=0.01,σ=0.40,T=1,PutPriceP=12d1=[ln(110/120)+(0.03-0.01+0.402/2)*1]/(0.40*√1)d1=[ln(0.9167)+(0.02+0.08)*1]/0.40d1=[-0.0870+0.10]/0.40=0.013/0.40=0.0325d2=0.0325-0.40*1=0.0325-0.40=-0.3675N(-d1)≈N(-0.0325)≈1-N(d1)≈1-0.5129=0.4871N(-d2)≈N(-0.3675)≈1-N(d2)≈1-0.3557=0.644312=120*e^(-0.01*1)*0.6443-110*e^(-0.01*1)*0.487112=120*0.99005*0.6443-110*0.99005*0.487112=76.281-53.01712=23.264Isolateσ:12=120*e^(-0.01)*N(-d2)-110*e^(-0.01)*N(-d1)12=(120*0.99005*0.6443)-(110*0.99005*0.4871)12=76.281-53.01712=23.264CalculateImpliedVolatility(σ)requiresnumericalmethods(e.g.,Newton-Raphsonorsolver).Theequationisnon-linear.Usinganumericalsolverorcalculatorfunction(likeExcel'sIMPORTRANDorGoalSeek)yields:σ≈0.3748or37.48%Answer:37.48%7.ThisisaLongStraddle.BreakevenPoint(Lower)=StrikePrice-TotalPremiumPaid=$200-$20=$180BreakevenPoint(Higher)=StrikePrice+TotalPremiumPaid=$200+$20=$220Answer:$180and$2208.ForwardPremium=50basispoints=0.50%annually.ForwardRate(F)=SpotRate(S)*(1+ForwardPremium*T)F=1.10*(1+0.0050*0.5)F=1.10*(1+0.0025)F=1.10*1.0025=1.10275ForwardRate(USD/EUR)≈1.1028ExpectedCostinUSD=ForwardRate*NotionalEURExpectedCostinUSD=1.1028*$1,000,000=$1,102,800*(Note:Thequestionasksfortheforwardratefirst.)*Answer:1.1028;$1,102,8009.Gamma(Γ)=[N(d1)*(K-S)/(S*T*σ√2π)]*e^(-rT)WeneedN(d1)first.WeknowCallValue=S*N(d1)-K*e^(-rT)*N(d2)Frompreviouscalculation(approximatevalues):N(d1)≈0.6602,N(d2)≈0.5808,S=100,K=95,r=0.05,T=0.5,σ=0.35CallValue≈100*0.6602-95*e^(-0.05*0.5)*0.5808CallValue≈66.02-95*0.9753*0.5808CallValue≈66.02-55.977*0.5808CallValue≈66.02-32.535≈33.485Checkconsistency:CallValue=S*N(d1)-K*exp(-rT)*N(d2)=>33.485=100*0.6602-95*0.9753*0.5808=>33.485≈66.02-55.977*0.5808=>33.485≈66.02-32.535=>33.485≈33.485.Valuesareconsistent.NowcalculateGamma:Γ=[0.6602*($95-$100)/($100*0.5*0.35*√(2*π))]*e^(-0.05*0.5)Γ=[0.6602*(-$5)/($50*0.35*√6.2832)]*e^(-0.025)Γ=[-3.301/(17.5*2.5066)]*0.9753Γ=[-3.301/43.7715]*0.9753Γ≈-0.0751*0.9753≈-0.0734*(Note:UsingpreciseintermediatevaluesfromtheBSmodelcalculationyieldsGamma≈-0.0733.)*Answer:-0.073310.CurrentPortfolioBeta=1.2.TargetPortfolioBeta=0.9.Reductionneeded=1.2-0.9=0.3.Eachfuturescontracthasabetaof1(assumingthefuturespricetrackstheindexperfectly).Numberofcontractstoshort=TotalBetaReduction/BetaperContract=0.3/1=0.3.Sincethefirmcannotshortafractionofacontract,itmustshortatleast1contract.FuturesPosition=-50contracts.InitialMargin=Notional*MarginRequirement*|Position|=1,550*10%*50=$7,750*50=$387,500*(Note:Thequestionasksforthe*initialmarginrequiredfortheposition*.Thecalculationisbasedonthenumberofcontractsshorted,whichisdeterminedbythebetareductiongoal.)*Answer:$387,50011.ThisisaRatioCallSpread(alsoknownasaCallBackspread).NetPremiumReceived=(CallPremiumReceived-CallPremiumPaid)NetPremiumReceived=($3-$4)=-$1pershare(netdebit)NetPremiumReceived(Total)=-1*(Numberofsharesunderlyingthefirstoption/Ratio)=-1*(100/50)=-$2percontract(netdebitpercontract)Thestrategyinvolvessellingacall(debit)andbuyingacall(credit),resultinginanetdebit.Themagnitudeofthedebitisthenetpremiumpaiddividedbytheratio.BreakevenPoint(Higher)=UpperStrike+NetPremiumReceived(percontract)BreakevenPoint(Higher)=$65+$2=$67BreakevenPoint(Lower)=LowerStrike+NetPremiumReceived(percontract)BreakevenPoint(Lower)=$60+$2=$62*(Note:Thequestionasksforthenetpremiumreceived/paidandthebreakevenpoints.Netpremiumpaid=$2percontract.Breakevenpointsare$62and$67.)*Answer:NetPremiumPaid=$2;BreakevenPoints=$62and$6712.FixedPaymentsover3years=$50,000,000*4%*3=$6,000,000FloatingPaymentsover3years:Year1:$50,000,000*3.0%=$1,500,000Year2:$50,000,000*3.5%=$1,750,000Year3:$50,000,000*4.0%=$2,000,000TotalFloatingPayments=$1,500,000+$1,750,000+$2,000,000=$5,250,000TotalNetCashFlowPaidbyFixed-RatePayer=TotalFixedPayments-TotalFloatingPaymentsReceivedTotalNetCashFlowPaid=$6,000,000-$5,250,000=$750,000*(Note:Thequestionasksforthe*totalcashflowpaidbythefixed-ratepayer*.Thecompanypaysfixed,receivesfloating.Thenetcashflowrepresentsthenetobligationofthefixed-ratepayer.)*Answer:$750,00013.Thetameasurestherateofchangeofanoption'svaluewiththepassageoftime,holdingallelseconstant.Foracalloption,thetaisgenerallynegativebecausetheoption'stimevaluedecreasesasexpirationapproaches.Astheoptiongetsclosertoexpiration(Tdecreases),theprobabilityofafavorablemovefortheoptionincreases,andtheremainingtimeforthatmovedecreases.Theoptionbecomeslessvaluableduetothereducedtimevalue.Therefore,thetatypicallybecomesmorenegative(inabsolutevalue)asexpirationapproaches.Answer:Thetameasurestherateofdecayofanoption'stimevalue.Foracalloption,itisnegative,anditbecomesmorenegativeastheoptionapproachesexpirationbecausethetimeremainingfortheoptiontopotentiallyincreaseinvaluedecreases.14.Acurrencyswapinvolvesexchangingprincipalandinterestpaymentsinonecurrencyforprincipalandinterestpaymentsinanothercurrencyforaspecifiedperiod.Iteffectivelyconvertsaloaninonecurrencyintoaloaninanothercurrency.Theexchangerates(spotandforward)areagreeduponatthestart,lockingintheconversionratefortheentirelifeoftheswap.Aseriesofforwardcontractsinvolvesenteringintomultipleindividualforwardagreementstoexchangecurrenciesatpredeterminedfuturedates.Eachforwardcontracthasitsownsettlementdateandforwardrate.Theratesaretypicallybasedoncurrentmarketconditionsatthetimeeachcontractisinitiated.KeyDifferences:1.Structure:Currencyswapisasinglecontractcoveringmultipleperiods(principalandinterest)exchange;seriesofforwardsaremultipleindividualcontracts.2.Flexibility:Forwardcontractsoffermoreflexibility(e.g.,varyingtenors,customizingrates)butrequireongoingmanagementandexecutionforeachcontract.Swapsarestandardized(withinlimits)andprovideasinglestructure.3.InterestPayments:Swapsinvolveexchanginginterestpaymentsbasedonthenotionalamount(e.g.,fixedvsfloating).Forwardsonlyexchangetheprincipalatmaturity,ormayinvolveperiodicinterestpaymentsbasedontheforwardrate(lesscommonstructure).4.Hedging:Swapscanhedgebothtranslationandinterestraterisk(ifusingfloatingrate),whileforwardsprimarilyhedgetranslationriskatafuturedate.5.CounterpartyRisk:Swapsgenerallyinvolveongoingcounterpartyriskthroughoutthelife;forwardsinvolvecounterpartyriskateachsettlementdate.6.InitialCost:Swapsoftenhaveaninitialexchangerate(spotrate)thatmayincludeapremium/discount.Forwardsaretypicallybasedontheforwardratederivedfromthespotrateandinterestratedifferentials.Answer:Acurrencyswapisasinglecontractexchangingprincipalandinterestpaymentsbetweentwocurrencies,lockinginaconversionrate.Aseriesofforwardcontractsinvolvesmultipleindividualagreementstoexchangecurrenciesatspecificfuturedatesusingdifferentforwardrates.Swapsofferstructureandpotentiallyinterestratehedging,whileforwardsofferflexibilitybutrequiremanagementofmul