2025年FRMPartI量化分析模擬沖刺卷考試時(shí)間：______分鐘總分：______分姓名：______一、1.LetXbeanormallydistributedrandomvariablewithmean5andvariance4.WhatistheprobabilitythatXislessthan3?2.SupposeXandYaretwoindependentrandomvariableswithX~N(0,1)andY~N(2,3).Whatisthedistributionof2X+Y?3.Asampleof30observationsistakenfromapopulation.Thesamplemeanis50andthesamplestandarddeviationis10.Constructa95%confidenceintervalforthepopulationmean.4.Inahypothesistest,thenullhypothesisisH0:mu=100andthealternativehypothesisisH1:mu≠100.Asampleof25observationsistakenwithasamplemeanof95andasamplestandarddeviationof15.Calculatethep-valueforthistestifthedatafollowsat-distribution.5.ExplainthedifferencebetweenaTypeIerrorandaTypeIIerrorinthecontextofhypothesistesting.二、1.ConsiderageometricBrownianmotion(GBM)assetpriceprocessdefinedbytheSDEdS=mu*S*dt+sigma*S*dW,wheremuisthedriftrate,sigmaisthevolatility,andWisastandardBrownianmotion.DerivethesolutionforS(t)usingIt?'slemma.2.ExplaintheconceptofaMarkovchainwithaclearexample.Whatisthesignificanceofthetransitionprobabilitymatrix?3.Defineastationaryprocess.Giveanexampleofastationaryprocessandexplainwhyitisstationary.4.Aportfolioconsistsofthreeassetswiththefollowingweights,expectedreturns,andstandarddeviations:AssetA(weight0.5,expectedreturn10%,standarddeviation15%),AssetB(weight0.3,expectedreturn12%,standarddeviation20%),AssetC(weight0.2,expectedreturn8%,standarddeviation10%).AssumethecorrelationbetweenAssetAandAssetBis0.4,betweenAssetAandAssetCis-0.2,andbetweenAssetBandAssetCis0.6.Calculatetheexpectedreturnandvarianceoftheportfolio.5.DescribethedifferencebetweenahistoricalsimulationandaMonteCarlosimulationinthecontextofriskmanagement.三、1.AEuropeancalloptionhasastrikepriceofKandanexpirationdateofT.TheunderlyingassetpricefollowsageometricBrownianmotion.DerivetheBlack-Scholesformulaforthepriceofthecalloption.2.Explaintheconceptofput-callparity.Providetheput-callparityrelationshipforEuropeancallandputoptions.3.A1-yearEuropeanputoptiononanon-dividendpayingstockhasastrikepriceof$50.Thecurrentstockpriceis$55.Therisk-freeinterestrateis5%.UsingtheBlack-Scholesmodel,calculatethepriceoftheputoption.4.Explaintheroleof希臘字母(Greekletters)inoptionspricing.Definedelta,gamma,theta,vega,andrho.5.Describethedifferencesbetweenaforwardcontractandafuturescontract.Whatarethekeycharacteristicsofeach?四、1.Abankismanagingthemarketriskofitsinterestratesensitiveassetsandliabilities.ExplainhowVaR(ValueatRisk)canbeusedtomeasuremarketrisk.WhatarethelimitationsofVaR?2.DefineValueatRisk(VaR)andExpectedShortfall(ES).Howdotheydifferintermsofriskmeasurement?3.Acompanyisconsideringaprojectwiththefollowingcashflows:Year0:-$1,000,000,Year1:$500,000,Year2:$600,000,Year3:$700,000.Therequiredrateofreturnfortheprojectis10%.CalculatetheNetPresentValue(NPV)oftheproject.4.Explaintheconceptofcorrelationinthecontextofportfolioriskmanagement.Howdoescorrelationaffectthediversificationbenefitsofaportfolio?5.Describethedifferenttypesofriskmodelsusedinriskmanagement.Discusstheadvantagesanddisadvantagesofeachtype.試卷答案一、1.P(X<3)=P(Z<(3-5)/sqrt(4))=P(Z<-1)=0.1587(Usingstandardnormaldistributiontable)2.SinceXandYareindependentandnormallydistributed,2X+Yisalsonormallydistributed.ThemeanisE[2X+Y]=2*E[X]+E[Y]=0+2=2.ThevarianceisVar(2X+Y)=Var(2X)+Var(Y)=4*Var(X)+Var(Y)=4*1+3=7.Therefore,2X+Y~N(2,7).3.ThestandarderrorofthemeanisSE=s/sqrt(n)=10/sqrt(30)≈1.8257.Thecriticalvaluefora95%confidenceintervalwith29degreesoffreedom(t-distribution)isapproximately2.045.Theconfidenceintervalismean±criticalvalue*SE=50±2.045*1.8257≈(46.06,53.94).4.Theteststatisticist=(samplemean-hypothesizedmean)/(standarddeviation/sqrt(samplesize))=(95-100)/(15/sqrt(25))=-5/3≈-1.667.With24degreesoffreedom,thep-valueforatwo-tailedtestisapproximately2*P(t<-1.667)≈2*0.0524≈0.1048(Usingt-distributiontableorcalculator).5.ATypeIerroroccurswhenthenullhypothesisistruebutisrejected.ATypeIIerroroccurswhenthenullhypothesisisfalsebutisnotrejected.Inthiscontext,aTypeIerrorwouldmeanconcludingthatthepopulationmeanisnot100whenitactuallyis100.ATypeIIerrorwouldmeanconcludingthatthepopulationmeanis100whenitisactuallynot100.二、1.Letf(S,t)=S.Thendf=dS.It?'slemmastatesdf=mu*f*dt+sigma*f'*sqrt(dt)*dW+1/2*sigma^2*f''*dt.Applyingthis:dS=mu*S*dt+sigma*S*dW.ThesolutionisS(t)=S(0)*exp((mu-sigma^2/2)*t+sigma*W(t)).2.AMarkovchainisastochasticprocesswherethefuturestateonlydependsonthecurrentstateandnotonthepaststates.Example:Asimpleweathermodelwheretheweathertomorrowdependsonlyontoday'sweather(sunnyorrainy)andnotonyesterday'sorearlierdays.ThetransitionprobabilitymatrixP=[[p_sunny_sunny,p_rain_sunny],[p_sunny_rain,p_rain_rain]]givestheprobabilityoftransitioningfromonestatetoanother.3.Astationaryprocessisastochasticprocesswhosestatisticalproperties(mean,variance,autocovariance)donotchangeovertime.Example:Whitenoise(aprocesswithconstantmean0,constantvariance,andzeroautocovarianceatalltimelags).Itisstationarybecauseitspropertiesareidenticalregardlessofwhenyouobserveit.4.E[Portfolio]=w_A*E[A]+w_B*E[B]+w_C*E[C]=0.5*10%+0.3*12%+0.2*8%=5%+3.6%+1.6%=10.2%.Cov(A,B)=rho*sigma_A*sigma_B=0.4*15%*20%=1.2%.Cov(A,C)=rho*sigma_A*sigma_C=-0.2*15%*10%=-0.3%.Cov(B,C)=rho*sigma_B*sigma_C=0.6*20%*10%=1.2%.Var(Portfolio)=w_A^2*Var(A)+w_B^2*Var(B)+w_C^2*Var(C)+2*w_A*w_B*Cov(A,B)+2*w_A*w_C*Cov(A,C)+2*w_B*w_C*Cov(B,C)=(0.5^2*15%^2)+(0.3^2*20%^2)+(0.2^2*10%^2)+2*0.5*0.3*1.2%+2*0.5*0.2*(-0.3%)+2*0.3*0.2*1.2%=0.5625%+0.36%+0.04%+0.36%-0.06%+0.14%=1.4125%.5.Historicalsimulationusespastmarketdatatoestimatethedistributionofpotentialfuturelosses.Itinvolvescalculatingtheactualhistoricalreturnsorlossesandthensimulatingfutureportfoliovaluesbasedonthesehistoricalscenarios.MonteCarlosimulationusesrandomnumbergenerationtosimulatemanypossiblefuturemarketscenariosbasedonassumedprobabilitydistributionsforreturns,volatility,etc.,andthencalculatestheportfoliovalueandlossdistributionforeachscenario.三、1.Letf(K,T,S,r,sigma)betheoptionprice.UsingIt?'slemmaforf,df=(partialf/partialK)*dK+(partialf/partialT)*dT+(1/2)*(partial^2f/partialS^2)*dS^2+(partialf/partialr)*dr+(partialf/partialsigma)*d(sigma).WeknowdS=r*S*dt+sigma*S*dW.Thedriftterm(r-sigma^2/2)*S*dtappearsintheoptionpriceduetothechangeinthepresentvalueofthestrikepriceK*exp(-rT).Thevolatilitytermsigma*S*dWcontributestheoptiontimevaluethroughthedeltahedges.2.Put-callparityisarelationshipbetweenthepriceofaEuropeancalloption,aEuropeanputoption,theunderlyingassetprice,thestrikeprice,andtherisk-freeinterestrate.Foranon-dividendpayingasset,itis:C+K*exp(-rT)=P+S.WhereCisthecallprice,Pistheputprice,Sisthecurrentstockprice,Kisthestrikeprice,ristherisk-freerate,andTisthetimetoexpiration.3.UsingtheBlack-Scholesformulaforaputoption:P=K*exp(-rT)*N(-d2)-S*N(-d1)whered1=(ln(S/K)+(r+sigma^2/2)*T)/(sigma*sqrt(T))andd2=d1-sigma*sqrt(T).WithS=55,K=50,r=0.05,T=1:d1=(ln(55/50)+(0.05+0.5^2/2)*1)/(0.5*sqrt(1))=(ln(1.1)+0.275)/0.5≈(0.09531+0.275)/0.5≈0.7706.d2=0.7706-0.5*1=0.2706.N(-d1)≈N(-0.7706)≈1-N(d1)≈1-0.7794=0.2206.N(-d2)≈N(-0.2706)≈1-N(d2)≈1-0.6098=0.3902.P=50*exp(-0.05*1)*0.3902-55*0.2206≈50*0.9512*0.3902-12.083≈18.602-12.083≈6.519.4.Greeklettersarepartialderivativesoftheoptionpricewithrespecttotheunderlyingvariables.Delta(δ)measurestherateofchangeoftheoptionpricewithrespecttotheunderlyingassetprice(delta=partialC/partialS).Gamma(γ)measurestherateofchangeofdeltawithrespecttotheunderlyingassetprice(gamma=partialdelta/partialS).Theta(θ)measurestherateofchangeoftheoptionpricewithrespecttotime(theta=partialC/partialT),representingthetimedecay.Vega(ν)measurestherateofchangeoftheoptionpricewithrespecttovolatility(vega=partialC/partialsigma).Rho(ρ)measurestherateofchangeoftheoptionpricewithrespecttotherisk-freeinterestrate(rho=partialC/partialr).5.Aforwardcontractisacustomized,privateagreementbetweentwopartiestobuyorsellanassetatapredeterminedpriceonafuturedate.Itdoesnotrequireaninitialmarginordailymarkingtomarket.Afuturescontractisastandardized,exchange-tradedcontracttobuyorsellanassetatapredeterminedpriceonafuturedate.Futurescontractsrequireinitialmargindepositsandaremarkedtomarketdaily,meaninggainsandlossesaresettledeachday.Futuresaremoreliquidandhavelowertransactioncoststhanforwardsbutinvolvedailysettlementrisk.四、1.VaRmeasuresthepotentiallossinvalueofaportfoliooveradefinedperiodforagivenconfidenceinterval.Forexample,a1-day95%VaRof$1millionmeansthatthereisa95%probabilitythattheportfoliowilllosenomorethan$1millionoverthenextday.VaRisusedforsettingrisklimitsandcapitalrequirements.Limitations:VaRdoesnotmeasurethemagnitudeoflossesbeyondtheVaRthreshold(tailrisk),itassumesreturnsarenormallydistributed(whichmaynotbetrue),anditcanbemanipulatedbychoosingappropriatetimehorizonsandconfidencelevels.2.VaRisastatisticalmeasureofthepotentiallossinaportfoliovalueatagivenconfidenceleveloveraspecificperiod.ExpectedShortfall(ES),alsoknownasConditionalValueatRisk(CVaR),istheexpectedlossgiventhatalossisoccurringatorbeyondtheVaRlevel.ESprovidesamorecomprehensivemeasureoftailriskthanVaRbecauseitconsiderstheaveragelossintheworst-casescenarios.WhileVaRtellsyouthemaximumlossyouareunlikelytoexceedwithhighconfidence,EStellsyoutheexpectedamountoflosswhenyoudoexceedthatVaRlevel.3.NPV=-1,000,000+500,000/(1+0.10)^1+600,000/(1+0.10)^2+700,000/(1+0.10)^3=-1,000,000+500,000/1.1+600,000/1.21+700,000/1.331=-1,000,000+454,545.45+495,867.77+525,920.78=-1,000,000+1,476,334.00=476,334.00.4.Correlationmeasuresthestatisticalrelationshipbetweenthereturnsoftwoassets.Itrangesfrom-1(perfectnegativecorrelation)to+1(perfectpositivecorrel