金融风险管理师(FRM)2012考试真题.doc

QuestionbankMonteCarloMethodsLetNbeannx1vectorofindependentdrawsfromastandardnormaldistribution,andletVbeacovariancematrixofmarkettime-seriesdata.Then,ifLisadiagonalmatrixoftheeigenvaluesofV,EisamatrixoftheeigenvectorsofV,andCCistheCholeskyfactorizationofV,whichofthefollowingwouldgenerateanormallydistributedrandomvectorwithmeanzeroandcovariancematrixVtobeusedinaMonteCarlosimulation?NC'CNNCELECannotbedeterminedfromdatagivenConsiderastockthatpaysnodividends,hasavolatilityof25%paandanexpectedreturnof13%pa.ThecurrentstockpriceisS0=$30.ThisimpliesthemodelSt+1=St(1+0.13At+0.25yAte),whereeisastandardnormalrandomvariable.Toimplementthissimulation,yougenerateapathofthestockpricebystartingatt=0,generatingasamplefore,updatingthestockpriceaccordingtothemodel,incrementingtby1andrepeatingthisprocessuntiltheendofthehorizonisreached.Whichofthefollowingstrategiesforgeneratingasampleforewillimplementthissimulationproperly?Generateasampleforebyusingtheinverseofthestandardnormalcumulativedistributionofasamplevaluedrawnfromauniformdistributionbetween0and1.Generateasampleforebysamplingfromanormaldistributionwithmean0.13andstandarddeviation0.25.Generateasampleforebyusingtheinverseofthestandardnormalcumulativedistributionofasamplevaluedrawnfromauniformdistributionbetween0and1.UseCholeskydecompositiontocorrelatethissamplewiththesamplefromtheprevioustimeinterval.Generateasampleforebysamplingfromanormaldistributionwithmean0.13andstandarddeviation0.25.UseCholeskydecompositiontocorrelatethissamplewiththesamplefromtheprevioustimeinterval.Continuingwiththepreviousquestion,youhaveimplementedthesimulationprocessdiscussedaboveusingatimeintervalAt=0.001,andyouareanalyzingthefollowingstockpricepathgeneratedbyyourimplementation.tSt—1eAS030.000.09300.03130.030.84930.21230.230.96170.23330.470.24600.06430.530.47690.12530.650.71410.18Giventhissample,whichofthefollowingsimulationstepsmostlikelycontainsanerror.CalculationtoupdatethestockpriceGenerationofrandomsamplevalueforeCalculationofthechangeinstockpriceduringeachperiodNoneoftheaboveInthegeometricBrownianmotionprocessforavariableS,I.Sisnormallydistributed.II.dln(S)isnormallydistributed.III.dS/Sisnormallydistributed.IV.Sislognormallydistributed.a.Ionlyb.II,III,andIVc.IVonlyd.IIIandIVConsiderthatastockpriceSthatfollowsageometricBrownianmotiondS=aSdt+bSdz,withbstrictlypositive.Whichofthefollowingstatementsisfalse?a.Ifthedriftaispositive,thepriceoneyearfromnowwillbeabovetoday’sprice.b.Theinstantaneousrateofreturnonthestockfollowsanormaldistribution.c.ThestockpriceSfollowsalognormaldistribution.d.Thismodeldoesnotimposemeanreversion.TheVasicekmodeldefinesarisk-neutralprocessforrwhichisdr=a(b−r)dt+σdz,wherea,b,andσareconstant,andrrepresentstherateofinterest.Fromthisequationwecanconcludethatthemodelisaa.MonteCarlo-typemodelb.Single-factorterm-structuremodelc.Two-factorterm-structuremodeld.DecisiontreemodelTheterma(b−r)inthepreviousquestionrepresentswhichterm?a.Gammab.Stochasticc.Reversiond.VegaWhichgroupofterm-structuremodelsdotheHo-Lee,Hull-White,andHeath,Jarrow,andMortonmodelsbelongto?a.No-arbitragemodelsb.Two-factormodelsc.Lognormalmodelsd.DeterministicmodelsAplausiblestochasticprocessfortheshort-termrateisoftenconsideredtobeonewheretherateispulledbacktosomelong-runaveragelevel.Whichoneofthefollowingterm-structuremodelsdoesnotincludethischaracteristic?a.TheVasicekmodelb.TheHo-Leemodelc.TheHull-Whitemodeld.TheCox-Ingersoll-RossmodelWhichofthefollowingstatementsaboutMonteCarlosimulationisfalse?a.MonteCarlosimulationcanbeusedwithalognormaldistribution.b.MonteCarlosimulationcangeneratedistributionsforportfoliosthatcontainonlylinearpositions.c.OnedrawbackofMonteCarlosimulationisthatitiscomputationallyveryintensive.d.Assumingtheunderlyingprocessisnormal,thestandarderrorresultingfromMonteCarlosimulationisinverselyrelatedtothesquarerootofthenumberoftrials.AriskmanagerhasbeenrequestedtoprovidesomeindicationofaccuracyofaMonteCarlosimulation.Using1,000replicationsofanormallydistributedvariableS,therelativeerrorintheone-day99%VARis5%.Undertheseconditions,a.Using1,000replicationsofalongoptionpositiononSshouldcreatealargerrelativeerror.b.Using10,000replicationsshouldcreatealargerrelativeerror.c.Usinganothersetof1,000replicationswillcreateanexactmeasureof5.0%forrelativeerror.d.Using1,000replicationsofashortoptionpositiononSshouldcreatealargerrelativeerror.ThemeasurementerrorinVAR,duetosamplingvariation,shouldbegreaterwitha.Moreobservationsandahighconfidencelevel(e.g.,99%)b.Fewerobservationsandahighconfidencelevelc.Moreobservationsandalowconfidencelevel(e.g.,95%)d.Fewerobservationsandalowconfidencelevelletthepdff(x)bepositiveatx=-1,0,1andzeroelsewhere.1.iff(0)=0.25,findE(x2)2.iff(0)=0.25andifE(X)=0.25,determinef(-1)andf(1)Onecardisdrawnfromastandard52carddeck.Indescribingtheoccurrenceoftwopossibleevents,anAceandaKing,thesetwoeventsaresaidtobe:(a)independent(b)mutuallyexclusive(c)randomvariables(d)randomlyindependent.WhichofthefollowingisNOTapossibleprobability?a.25/100b.1.25c.1d.0Amongtwenty-fivearticles,ninearedefe