FRM一級基礎(chǔ)段估值模型（打印版）

??:ZacWeightingsinFRMPart?1234?????????????????Bond.5.DiscountBasicMethodReplicationMethodBondP&LComponentsReviews??SpotRatezAt-periodspotrate,orzerorate,istheinterestrateearnedwhencashisreceivedatjustonefuturetime.zTherelationshipbetweenspotratesandmaturityiscalledthetermstructurespotrates.ForwardRatezForwardratesinterestratescorrespondingtoafutureperiodimpliedbythespotcurve.zAllforwardratescomputedusingspotrates,andallspotratescanbecomputedusingtheappropriateforwardrates.1+R??1+?,??????=1+R?R???R?Te?×e?=e???,?=??TDiscount?DiscountzThediscountd(t),foraterm(t)years,givesthepresentoneunitcurrency($1)bereceivedattheendthatterm.rtdt=1+dt=m0.5.51122.53Spot,ForwardRates???ForwardRateandSpotRate0110.94%F?.?,?21.37%21+1+=1+2F?.?,?=1.79%7BasicMethodHowdeterminethepriceabond??C?C?(1+S?)C?(1+y)P=++?+=?1+?(1+S?)?zSupposethata2-yearbondwithaprincipal$100providescouponsattherate6%perannumTheratescomputethebondprice:TreasuryZeroRatesMaturity(Years)ZeroRate(%)(continuouslyCompounded)0.55.011.5.02P=3e??.??×?.?+3e??.???×?+3e??.???×?.?+103e=98.398BasicMethodExampleThetablebelowshowsselectedT-bondpricesforsemiannualcoupon,$100facebonds.Calculatediscountfactorsforthesegivenbondprices.BondCouponMaturityPrice124%0.51101-236.3%104-22+4%2332Bond1:100+100××d0.5=101+2d0.5=0.99726.3%26.3%222.532Bond2:100××d0.5+100+100××d1=104+d1=0.9845BasicMethod???PricingBondusingDiscountFactors,SpotRates,orForwardRateszAssumea1-yeartreasurybondthata8%semi-annualcoupon.MaturitySpotRate(%)Discount6MonthForwardRate(%)0.501.001.502.002.500.941.371.822.513.080.9953030.9864780.9732140.9513360.9265470.941.792.734.585.37BasicMethodPricingBondusingDiscountFactors,SpotRates,orForwardRateszCalculatetheBondPriceusingDiscountPrice=$4×0.995303$104×0.986478=$106.57CalculatetheBondPriceusingSpotRates+z$04.94%2$104Price=+=$106.571.37%1+1+2zCalculatetheBondPriceusingForwardRates$04.94%$104×Price=+=$106.570.94%21.79%1+21+1+2ReplicationMethodLawoneprice:zAbsentconfoundingfactors(e.g.,financing,taxes,creditrisk),identicalsetscashflowsshouldsellforthesameprice.Whilethelawonepriceisintuitivelyreasonable,itsjustificationrestsonastrongerfoundation.Itturnsoutthatadeviationthelawonepriceimpliestheexistenceanarbitrageopportunity,thatis,atradethatgenerateswithoutanychancelosingzWhenmispricinghappens,arbitragemaynotoccurbecausefactorsotherthanpromisedcashflowsoccasionallyconsideredinthewaytheinstrumentspriced,suchastaxtreatmentandReplicationMethod?ExampleThreebondyieldsandpricesshownbelow?MaturityYTMCouponPrice(%par)95.2381231year2years2years5%6%6%0%0%6%99.00100The2-yearspotrateis6.2%.Isthereanarbitrageopportunityusingthesethreebonds?Ifso,describethetradesnecessaryexploitthearbitrageopportunity?ReplicationMethod?Example$10001.06Bond1.06Bond2$100$6$6+$1001Bond30.06B?+1.06B?=B?0.06×95.238+1.06×99.00=110.6543Bonds3isundervalued,sobuybond3.Bond1andBond2overvalued,thensellthemReplicationMethod?ExampleTime=01year2years(cost6%-1,000,000.00+60,000(coupon)+1,060,000(coupon)-60,000(maturity)couponbonds)(proceeds+57,142.800%couponbonds)(proceed%couponbonds)+1,049,400.00-1,060,0000+106,542.80Net00(maturity)BasicMethod?PriceanAnnuityzAnannuitywithsemiannualpaymentsisasecuritythatapaymentc/2everysixmonthsforTyearsbutneverafinal“principal”payment(i.e.,FV=0).ThepriceanA(T),isgivenby:Cy1A=1?y21+?PriceanPerpetuityzAperpetuitybondisabondthatcouponsThepriceaperpetuityissimplythecoupondividedbytheyield.Priceaperpetuity=c/yzBondReturn?GrossRealizedReturnsP+c?PR?=PzzIfwantlookatthereturnoveralongerperiod,mustconsidertheinvestmentthecouponas.Supposethattheinitialpurchasepriceabondis$98,andthepurchaseoccurredimmediatelyafteracouponpaymentdate.Itearns$1.75couponeverysixmonths.Ayearlaterthepriceis98.7.Assumingthatthecouponcanbeinvestedatanannualizedrate2.2%(semi-annualcompounding),thegrossrealizedreturnis:98.7?98+1.75+1.75×1+1.1%=0.043198BondReturn?zzIncorporatesfundingcostP+c?BR?=PContinuetheexample,assumingthatthefundsbuyfinancedat%perannum(semi-annualcompounding),thenetrealizedreturnis:8.7?98×1+1.5%?+1.75+1.75×1+1.1%39=0.012898BondReturnthecouponpaymentscanbereinvestedataninterestrateequaltotheyieldtomaturitythebondisheldmaturityIfthebondisnotheldtheinvestorfacestheriskthathemayhavesellforlessthanthepurchaseprice,resultingareturnthatislessthantheyieldknownasinterestraterisk.Reinvestmentriskexist.Futureinterestratescanbelessthantheyieldmaturityatthetimebondispurchased,knownasreinvestmentrisk.BondReturn?YieldMaturityzTheYTMabondisthesinglediscountrateatwhichallcashflowsthebonddiscountedandsummeduptheprice.Example:Supposeabond$40everysixmonthsforfouryearsandafinalpayment$1,000atmaturityinfouryears.Ifthepriceis$850,calculatetheYTM.z9Answer:TheYTMistheythatsolvesthefollowingequation:$40y2$40y2$40+$1000$850=++?+y21+1+1+FV=$1000;PV=-850;PMT=40;N=8??&37?,?<6.46?<7012.92%BondReturn?ExampleAbondwitha$100a4%couponannuallyfor3years.Thespotratescorrespondingthepaymentdatesasfollows:year1:4.5%;year2:5%;year3:5.5%.CalculatethepricethebondusingspotratesanddetermineintheYTMforthebond.zAnswer:Thepricethebondusingthespotratesisasfollows.44104P=++=96.02371.0451.05?1.055?zComputetheYTM:441041+YTM96.0237=++1+YTM1+YTMFV=$100;PV=-?????????307???1???&37?,?<?????<70?????BondReturn?RelationshipbetweenSpotRatesandYTM?+YTMCF?1+YTMCF?1+YTMP=++?+1?+R?CF?1+R?CF?1+R?=++?+1zzYTMisakindaverageallthespotrates.Whenisaflattermstructurespotrates,theyieldmustequalthosespotrates.zzWhenthetermstructurespotratesisupward-sloping,theyieldthetwo-yearbondwillbebelowthetwo-yearspotrate(usetwo-yearbondasexample).Whenthetermstructurespotratesisdownward-sloping,theyieldthetwo-yearbondwillbeabovethetwo-yearspotrate.BondReturnInterestRateInterestRate98765439876543ForwardCurveSpotCurveYieldCurveYieldCurveSpotCurveForwardCurve012345678910012345678910Upward-SlopingDownward-SlopingBondReturn?CouponEffectzThefactthatcorrectlypricedbondswiththesamematuritybutdifferentcouponshavedifferentyieldstomaturityiscalledthecouponeffect.zAsthecouponrises,theaveragetimeittakesbondholderstorecovertheircashflowsfalls.Therefore,thespotratesfortheearlypaymentdatesisbecomingmoreimportantindeterminingtheyieldtomaturity.領(lǐng)取考前題唯一微：xuebajun888s99upward-slopingtrend,theyieldmaturityfallsasthecouponrises.downward-slopingtrend,theyieldmaturityrisesasthecouponrises.BondReturn???azThepriceasecurityisbydiscountingacashflowsusinganappropriatetermstructureplusaspread.ccP=++?1+f1+s1+f1+s1+f2+sc+F+1+f1+s1+f2+s…1+fT+s2P&LComponentsDecompositionP&Lz99zTheandlossconsistsboth:Priceappreciation(ordepreciation);a.k.a.,capitalgainorloss.Cash-carry:cashflowssuchascouponpayments.Priceappreciationcanbedecomposedintothreecomponents:?Carry-roll-down:thepricechangeduethepassagetimeratesmoveasexpectedbutwithnochangeinthespread.?Themostcommonassumptionwhenthecarryroll-downiscalculatedisthatforwardratesrealized(i.e.,theforwardrateforafutureperiodremainsunchangedasmovethroughtime).P&LComponentsDecompositionP&L?Ratechange:thepriceeffectrateschangingtheintermediatetermstructuretothetermstructurethatactuallyattimet+1.?Thepriceappreciationduetoaspreadchangeisthepriceeffectduetotheindividualspreadchangings(t)tos(t+1).P&LComponents?ExampleStartperiod201020112012priceP&L1-1Pricingdate:2010-1-1;annualcoupon=1Initialstructure2%3%4%forwardsspreads0.5%0.5%0.5%Pricingdate:2011-1-1;annualcoupon=1Carry-roll-structure3%4%0.5%0.5%2%3%0.5%0.5%1-11-193.0229+1.32569994.34856.18005.2577downspreadsstructurespreadsCarry-Roll-Down:2.3256Rate+1.8315changeSpreadstructurechangespreads2%1%3%1%-0.9223P&LComponents11101P=+++=93.02291+2.5%1+2.5%1+3.5%1+2.5%1+3.5%1+4.5%1101P???????????????==94.3485=96.181+3.5%1+3.5%1+4.5%1101P??????????=+1+2.5%11+2.5%1+3.5%101P????????????=+=95.25771+3%1+3%1+4%Exercise1?Thepriceofathree-yearzero-coupongovernmentbondis85.16.Thepriceofasimilarfour-yearbondis79.81.Whatistheone-yearimpliedforwardratefromyear3toyear4?A.5.4%B.5.5%C.5.8%D.6.7%??Answer:D領(lǐng)取考前：xuebajun888s題唯一微信Exercise2?Thefollowingdiscountfunctioncontainssemi-annualdiscountfactorsouttwoyears:d(0.5)=0.9970,d(1.0)=0.9911,d(1.5)=0.9809,d(2.0)0.9706.Whatistheimpliedeighteen-month(1.5year)spotrate(aka,.5yearrate)?=1A.0.600%B.1.176%C.1.290%D.1.505%?Answer:CExercise3?Anannuity$10everyyearfor100yearsandcurrentlycosts$100.TheYTMisclosestto:A.5%B.7%C.9%D.10%?CorrectAnswer:DExercise4?A$1,000bondcarriesacouponrate10%,couponsandhas13yearsremainingtoMarketratescurrently9.25%.Thepricethebondisclosestto:A.$586.60B.$1,036.03C.$1,055.41D.$1,056.05?CorrectAnswer:DExercise5?Abondportfoliomanagerinvests$20millioninabondissuedatparthatmaturesin30years,andwhichpromisestopayanannualinterestrateof9%.Theinterestispaidonceperandthepaymentsarereinvestedatanannualinterestrateof8%.Thefirstpaymentisoneyearfromtoday.Whatistheannualyieldonthisinvestment?A.8.185%B.8.285%C.8.385%D.8.415%??Answer:C領(lǐng)取考前：xuebajun888s題唯一微信Bond12..ShiftsNon-ParallelShiftsParallelStructureShifts??One-FactorAssumptionzTheonlysignificantassumptionmadeabouthowthetermstructurechangesisthatallratechangesdrivenbyonefactoThesimplestone-factormodelassumesthatallratemovebythesameamount.DurationAnalysiszYielddurationmeasuresthesensitivityapricetoitsyieldchange.9Plainbond:?MacaulayDuration&ModifiedDuration&DollarDuration&DV01?Convexity9Embeddedoption:?Effectiveduration&EffectiveconvexityParallelStructureShifts??MacaulayDurationzperiodcashflowreturningweightedbydiscountedcashPVCF?PMac.D=?×t=?(w?×t)ExampleIfabondhasapresentUSD93.06withacashflowinoneyearprovidingapresentUSD5.45andacashflowintwoyearsprovidingapresentUSD87.60,theMacaulaydurationwouldbe:5.4587.60×1+×2=1.941493.0693.06zzForaplainbond,theMacaulaydurationislessthanorequaltoitsForazerocouponbond,theMacaulaydurationequalstoitsParallelStructureShifts?ModifiedDurationandDollarDurationz==1+z=×P×Pzz=.=×P=yParallelStructureShifts?DV01zTheDV01istheabsolutevaluethepricechangeabondonepointchangeinyield.??DV01=???DV01=MD×BondValue×0.0001?Examplezz9Macaulayduration=1.9414Bondvalue=93.06Thisindicatesthata1bpschangeinthecontinuouslycompoundedyieldtogiverisetoapricechangeof:?1.9414×93.06×0.0001=?0.01807ParallelStructureShifts?z9aaAAaBBF×F=aaaaz1===zzParallelStructureShifts?Price-YieldrelationshipParallelStructureShifts??LimitationsDurationzDurationprovidesagoodapproximationwhensmallparallelshiftintheinterestratetermstructure.itwillprovideapoorapproximationifnon-parallelshiftorthechangeis.ConvexityzAmeasurethenon-linearrelationship.Convexity=?(w?×t?)zWhenratesexpressedwithcompoundingmtimesperyear:ConvexityModifiedConvexity=y1+mRiskMetrics???PriceApproximation1Py?+?y=Py?+f?y??y+f??y??y?+?2DollarDollarDurationConvexityzzzTheactual,exactprice:P=f(y?+?y)Thedurationestimate:P=P?D×P×?yThedurationandconvexityestimate:1P=P?D×P×?y+×C×P×?y2ParallelStructureShiftsDurationandConvexityAnalysiszTheeffectparallelshiftsinterestratetermstructurecanbeaccuratebyaddingconvexityanalysistheanalysisduration.zIfCistheapproximatepricechangecanberefinedto:1??=?????+CP??2zThisapproximationallowsrelativelyparallelshiftstobeconsidered.ParallelStructureShiftsExampleConsidera10-yearzero-couponbondtradingatayield6%,semiannuallycompounding.Thepresentthebond:P=100/1+6%/2??=55.368CalculationDurations,forazero-couponbond:zzzMacaulayduration=10yearsModifiedduration:MD=10/1+6%/2=9.708Dollarduration:DD=MD×P=9.708×55.368=537.553ParallelStructureShifts???Example?1?canusethedurationtocalculatethebondpriceapproximatelywhentheyieldchanges,forexample,yieldgoesto7%6%TheexactpricebondisP=100/1+7%/2??=50.257UsingdurationtoapproximateP=?+f?y???=55.368?537.553×1%=$49.993CalculateconvexityConvexity=141××21×20=98.9736%1+2Usingdurationandconvexitytoapproximate1P=55.368?537.553×1%+×98.973×55.368×1%?=50.2672ParallelStructureShiftsExample?2?Ifyieldgoes5%6%TheexactpricebondisP=100/1+5%/2??=61.027UsingdurationtoapproximateP=?+f?y???=55.368?537.553×??1%?=$60.744Usingdurationandconvexitytoapproximate1P=55.368?537.553×?1%+×98.973×55.368×?1%2=61.017ParallelStructureShiftsNegativeConvexityzAcallablebondgivestheissuertherighttoallorpartthebondbeforethespecifiedmaturitydate.zMostmortgagebondsnegativelyconvex,andcallablebondsusuallyexhibitnegativeconvexityatloweryields.1ParallelStructureShifts?EffectiveDurationandEffectiveConvexityzInabond(withoutembeddedoptions)cantypicallyusemodifiedandeffectiveWhenthebondcontainsembeddedoptions,prefereffectiveduration:??/PP?PD=?=?????999zP=initialobservedbondprice??=changeinrequiredyield(indecimalform)Effectiveconvexity:anapproximatemeasureconvexityD??D???P+??2P??y?C==ParallelStructureShifts?EffectiveDurationandEffectiveConvexityzExample:Supposethereisa10-yearbondwithanannualcouponof6%tradingatComputethebond’sdurationfora40basispointincreaseanddecreaseinyield,andcomputetheconvexityofthisbond.z9Answer領(lǐng)取考前：xuebajun888s題唯一微信Ifinterestratesriseby40basispointsN=10;PMT=6;FV=100;I/Y=6.4;CPT?PV=?97.11Ifinterestratesfallby40points9N=10;PMT=6;FV=100;I/Y=5.6;CPT?PV=?103103?97.11D=C==7.36252×100×0.004103+97.11?2×100=68.75100×0.004?ParallelStructureShifts?HedgingbasedonEffectiveDurationz9999zSupposeTheeffectivedurationaninvestmentis?ThetheinvestmentisVThedurationabondis?ThethebondisPIfhedgeagainstsmallparallelshift,thenthepositioninthebondis:VD?D?P=ParallelStructureShifts???HedgingbasedonEffectiveDurationandConvexityzcanusetwobondsbotheffectivedurationandeffectiveconvexityzero.Suppose999P,D,andCthevalue,duration,andconvexitythefirstbondP,D,andCthevalue,duration,andconvexitythesecondbondD?,andC?thevalue,duration,andconvexitythepositionthatistobehedged.zcanbothdurationandconvexitybychoosing?and?:?VD??????D?=0VC?+??+?C?=0ParallelStructureShiftsPortfolioDurationandConvexityzInbothmodified(effective)durationandportfoliodurationandconvexityequaltheweightedsumindividual,durationsandconvexitieswhereeachweightisitsasapercentageportfoliozExampleCouponMaturityYTMDC567.00%.00%.00%54.00%22.97%4.4122.9215305.00%32.37%10.11132.545.50%44.66%14.00299.36D=(0.2297×4.41)+(0.3237×10.11)+(0.4466×14)=10.54C=(0.2297×22.92)+(0.3237×132.54)+(0.4466×299.36)=181.86ParallelStructureShiftsBulletversusBarbellPortfoliozAmanagerpurchase$100millionBatacost$100,000,000.Thepaymentssemi-annual.UsingAandCconstructaportfoliowiththesamecostandduration.ABC2465?+?=100,000,000??00,000,000×4.7060+×14.9120=8.17551100,000,000?=66m,?=34mC?????????=0.66×25.16+0.34×331.73=129.4?Exercise1?Abondhasacouponrateof6%perannum(thecouponsarepaidsemiannually)andasemiannuallycompoundedyieldof4%perannum.Thebondmaturesin18monthsandthenextcouponwillbepaid6monthsfromnow.Whichnumberbelowisclosesttothebond’sMacaulayduration?領(lǐng)取考前題唯一微：xuebajun888sA.1.023yearsB.1.457yearsC.1.500yearsD.2.915years?CorrectAnswer:BExercise2?Azero-couponbondwithamaturity10yearshasanannualeffectiveyield10%.Whatistheclosestforitsmodifiedduration?A.9B.10C.99D.100?Answer:AExercise3?Themodifieddurationis10.46abondwithacurrentprice$716.38.WhatistheDV01?A.$0.40B.$0.75C.$1.25D.Needinformation(yield,maturity)?Answer:BExercise4?Abondportfoliohasthefollowingcompositions:zzzPortfolioA:price$90,000,modifiedduration2.5,longpositionin8bonds;PortfolioB:price$110,000,modifiedduration3,shortpositionin6bonds;PortfolioC:price$120,000,modifiedduration3.3,longpositionin12bonds;Allinterestrates10%.Iftheratesriseby25bps,thenthebondportfoliowillA.Decreaseby$11,430B.Decreaseby$21,330C.Decreaseby$12,573D.Decreaseby$23,463CorrectAnswer:AExercise5?A8%semiannualcouponbondwith$100currentlytradesat$78.75andhasaneffectiveduration9.8yearsandaconvexity130.Whatisthepricethebondiftheyieldfallsby150basispoints?A.$67.17B.$86.47C.$91.48D.$95.43?CorrectAnswer:CExercise6?Whichofthefollowingstatementsisalwayscorrect?A.Nomethodcanhedgeinterestraterisk.B.Single-factormodelassumemean-reversionbetweenoneshort-termandonelong-termrate.C.Single-factormodelsassumerisk-freesecuritieshavecreditexposure.D.Single-factormodelsassumethatallratechangesaredrivenbyone領(lǐng)取考前押題唯一微：xuebajun888s?Answer:DNon-ParallelStructureShifts?ModelingNon-ParallelStructureShiftszInpractice,theremanydifferenttypesnon-parallelshifts.Sometimesshort-termratesmovedownwhilelong-termmoveup,orviceversa.Occasionally,shortlong-terminterestratesmoveinonedirection,whilemedium-termratesmoveintheotherdirectionyNonparallelTNon-ParallelStructureShifts?PrincipalComponentsAnalysiszAstatisticaltechniquecalledprincipalcomponentsanalysiscanbeusedtounderstandtermstructurechangesinhistoricaldata.Principalcomponentanalysiscananalyzetheeffectsofmultiplefactorsandestimatetheirrelativeimportanceindescribingmovementsinthetermstructure.zzThistechniquelooksatthedailychangesininterestratescorrespondingtovariousmaturitiesandidentifiesfactorsthathavethefollowingcharacteristics:領(lǐng)取考前題唯一微：xuebajun888s?Thesefactorsuncorrelated;?Dailychangesintermstructurelinearcombinationsthefactors;?Thefirsttwoorthreefactorsaccountforthemajoritytheobserveddailymovements.Non-ParallelStructureShifts????zz?Non-ParallelStructureShifts?Forourdata,thetotalvarianceis14.15?+4.91?+?+0.68?=235.77zzThefirstfactoraccountsfor84.9%(=14.15?/235.77)Thefirsttwofactorsaccountfor:14.15?+4.91?=95.14%235.77zThefirstthreefactorsaccountfor97.66%thevariance.14.15?+4.91?+2.44?=97.66%235.77Non-ParallelStructureShifts?Key-RateExposurez9Key-RateShiftsAssumptionThecrucialassumptiontherateisthatallratescanbedeterminedasafunctionofarelativelysmallnumberofkeyrates.Afewratesalongtheterm-structurearepickedwhicharerepresentativeofthecurve.領(lǐng)取考前題唯一微：xuebajun888s99Therateagivenmaturityisaffectedsolelybyitsclosestkey-rate.Shiftsinthekey-ratesdeclinelinearly.1056Non-ParallelStructureShifts?Key-RateExposurez99Key-RateShiftsMetricsRate’01s:whichistherateequivalentDV01.RateDuration:whichistherateequivalentdurations.?P?y1P?P?yDV01???=?0.0001×D???=??zExample:Calculate30-yearrate01andratedurationapplyingaonebasispointshift.=0.1033=41.129425.01254?25.11584KeyRate01??=?0.0001×0.01%5.01254?25.115842KeyRateDuration??=?25.11584×0.01%Non-ParallelStructureShifts???Example1SupposeaportfolioconsistsaUSD1millioninvestmentineachaone-and15-yearcouponbond.Thedecreaseintheforaone-basis-pointincreaseinthespotratesisasfollows:Decreaseinfora1bpsIncreaseinSpotRatesSpotRateMaturity135915PortfolioDecrease96.72270.26433.85677.93957.66Supposefive-yearandten-yearinterestrateselectedasratetobeusedinanalysis,then:6Non-ParallelStructureShiftsExample1ChangesinSpotRateforChangesinRateSpotRateMaturityRateShift1100350109150Five-year0.66670.3333000.20.801KR01sforPortfolioCalculationPartial01?276.9096.72+(0.6667×270.26)KR01?0.3333×270.26+433.85+(0.2×677.93)659.520.8×677.93+957.661,500.00KR01?Non-ParallelStructureShiftsExample2Supposethreedifferenthedginginstrumentsusedtohedgerisks.TheKR01sportfolioandhedginginstrumentsisshowninthefollowingtable:DataforHedgingUsingKR01sPortfolioHedgingInstruments1402636KR011KR012KR0132524767704448825024752+40x?+6x?+6x?=076+4x?+44x?+8x?=070+2x?+8x?+50x?=0x?=?3,x?=?8,x?=?14Non-ParallelStructureShifts?ForwardBucketShiftzEachforward01iscomputedbyshiftingtheforwardratesinthatbucketbyonebasispoint.Non-ParallelStructureShifts?EstimatingPortfoliozzzRegulatorsBankstoconsidertendifferentKR01swhenanalyzingtherisktheirportfolios(including3-month,6-month,and30-yearspotrates).BanksneedalltheKR01szero,buttheyneeduseKR01exposureandstandarddeviationthetenratestoestimateriskmeasuressuchasorExpectedShortfall.Theformulaforthestandarddeviationthechangeintheportfolioinonedayis:??=?????????×KR01?×KR01???isthevolatilitythedailymovementinratei(measuredinbps)and???isthecorrelationbetweenthedailymovementsinrateiandj.Exercise?Assumeyouownasecuritywitha2-yearrateexposure$4.78,andyouwouldtohedgeyourpositionwithasecuritythathasacorresponding2-yearrateexposure0.67per$100facevalue.Whatamountfacewouldbeusedhedgethe2-yearexposure?A.$478B.$239C.$713D.$670?Answer:COption1.2.3.OtherAssetsOne-Step?Risk-Neutralzdefinerisk-neutralworldasoneinvestorsdonotadjusttheirexpectedreturnbasedrisk,sotheexpectedreturnallassetsisrisk-freeinterestrate.zTherisk-neutralvaluationprinciplestatesthatifassumeinarisk-neutralworld,cangetafairpriceforaderivative.One-Step?ExampleThestockpriceABCcompanyis$20Thestockwillgoup2ordown18threemonthsWhattheEuropeancallpricez2thisstockmonthsnow?SupposethestrikepriceisK=$21,continuouslycompoundedrisk-freerateis12%.Price=$22pOptionPrice=$1Price=$20OptionPricef1–pPrice=$18OptionPrice=$0One-Step?Solution1zzzzzLong?stock,short1call.Inarisk-neutralworld:22?–1=18??=0.25att0=20?–fatt1=22?–120×0.25?fe?.??×?.??=22×0.25?1zf=0.633?Solution222P+181?P=20eP=0.6523f=1×p+0×1?pe??.??×?.??=0.633One-Step?Generalizationupu=u–K,=0f1–pdd=d–K,zzLong?stocksandshort1calloptionhedgetherisk.S????f?=S???f,eliminaterisksinbothcases.f?f?S?u?S?d?=f=pf?+1?pf?ee?du?dp=u=ed=eMulti-Step?EuropeanOptionsS0uufS0SfS0udfudfS0ddfe?du?dp=f=ep?f??+2p1?pf??+1?p?f??Multi-Step???Example:EuropeanCallOptionwithUpandDownInformedbyAssetTimeRisklessYield5%2%$810$8000.520%udp1.10520.90480.5126$989.34$895.19732.92$810$810$$663.17Multi-StepExample:EuropeanCallOptionwithUpandDownInformedbyu=e=e??%×?.??=1.1052d=0.9048e(???)???du?de?0.90481.1052?0.9048p===0.5126189.34100.66.0653.41050(189.34×0.5126+10×1?0.5126)???.??×?.??=100.6610×0.5126+0×1?0.5126???.??×?.??=5.06100.66×0.5126+5.06×1?0.5126???.??×?.??=53.4Multi-StepAmericanOptions0ztheoptionateachnodeisnolessthantheintrinsicvalue.Sf0udf0Multi-Step?Example:AmericanPutOptionwithpricejump+/-20%AssetTime50$52RisklessYield5%0%$2udp282$72$6040$4832$50$$Multi-Step?Example:AmericanPutOptionwithpricejump+/-20%01.4147/05.09/249.4634/12200×0.6282+4×1?0.6282e??.??×?=1.4147×0.6282+20×1?0.6282e??.??×?=9.4634.4147×0.6282+12×1?0.6282e??.??×?=5.0941OtherAssets??OptionsonwithDividendse?dp=Everythingelseabouttheisthesameasbefore.OptionsonIndicesu?dzzindexprovidesadividendyield.thevaluationshouldinvolvethemodificationasabove.??OptionsonCurrenciesCurrencycanbeconsideredasanassetprovidingayield.OptionsonFutureszzItcostsnotingintoafuturescontractandcanacontractastockpayingadividendyieldTherefore,get:1?dp=u?dMulti-Step?IncreasingtheNumberTimePeriodszAsthenumbertimestepsisincreased(sothat?Wbecomessmaller),thebinomialmodelmakesthesameassumptionsaboutstockpricebehaviorastheBlack-Scholes-Mertonmodel.WhenthebinomialisusedpriceaEuropeanoption,thepriceconvergestheBlack-Scholes-Mertonprice,asexpected,asthenumbertimestepsisincreased.Exercise1?Astockcurrentlytradesat$10.theendthreemonths,thestockwilleitherbe$11or$9.Thecontinuouslycompoundedrisk-freerateinterestis3.5%perThea3-monthEuropeancalloptionwithastrikeprice$10isclosestto:A.$0.11B.$0.54C.$0.65D.$1.01?Answer:BExercise2?AtraderhasanAmericanputoptionwithstrikeprice$50.Theunderlyingassetisstockwithaspotprice$40.Usinganone-stepbinomialtoevaluatetheoption.Supposethestockpricewillgoupordownby$8in6month,therisk-freerateis6.2%,whatisthethisAmericanA.USD8.19B.USD8.45C.USD10.00D.USD10.32?Answer:COptionModelAssumptions?AssumptionszzThestockpricefollowstheprocesswith?and?constant.notransactioncostsortaxes.Allsecuritiesperfectlydivisible.zzzzzTherenodividendsduringthelifethederivative.Therenorisklessarbitrageopportunities.Securitytradingiscontinuous.Therisk-freerateinterest,isconstantandthesameforallmaturities.Theoptionsbeingconsideredcannotbeexercised.PriceMovement?LognormalAssumptionzzGeometricBrownianmotiondS?=uSdt+?SdZLemma?f?f12??f?fdft,S=+uS?+??S?dt+?S?dZ??t?S?S??SzzAsthestockpricefollowsaGeometricBrownianMotion,ifdefineG=lnS,usinglemma,have:1dG=u???dt+?dZ2Avariablehasalognormaldistributionifthenaturallogarithmthevariableisnormallydistributed.WhiletheequationaboveshowsthatlnS?isnormallydistributed.Thestockpriceislognormallydistributed.PricingFormulas??B-SdifferentialequationV=eEmaxS??K,0ThePriceEuropeanOptionscall=S?N??KeNd?put=KeN?d??S?N????2lnS?/K+r±T?,?=?TzzzN?isthedeltaofcallNd?istheprobabilityofcallexercise1?Nd?istheprobabilityofputexercisePricingFormulas?ExamplezzzzzEuropeancalloptionpriceis$10,is$9.is20%.issixmonths.Risklessrateis5%.20%?2ln10/9ln10/9+25%+×0.5×0.5?==0.992=0.8510%×0.50%2+5%?2d?=20%×0.5call=10N0.992?9e??.??×?.?N0.851=1.35OtherAssets?OptionsonwithDividendsc=S?eN??KeNd?p=KeN?d??S?eN??lnS?/K+r?q±??/2T?,?=?T??OptionsonCurrencieszBehavesastockpayingadividendyieldattheforeignrisk-freerate.OptionsonFutureszBehavesastockpayingadividendyie