FRM一级基础段估值模型（打印版）

䇢ᐸ:ZacWeightingsinFRMPartĉ1234¾••¾¾¾•••••••••¾¾Bond.5.DiscountBasicMethodReplicationMethodBondP&LComponentsReviews¾¾SpotRatezAt-periodspotrate,orzerorate,istheinterestrateearnedwhencashisreceivedatjustonefuturetime.zTherelationshipbetweenspotratesandmaturityiscalledthetermstructurespotrates.ForwardRatezForwardratesinterestratescorrespondingtoafutureperiodimpliedbythespotcurve.zAllforwardratescomputedusingspotrates,andallspotratescanbecomputedusingtheappropriateforwardrates.1+Rଵభ1+ଵ,ଶ୘మି୘భ=1+RଶRଶଶെRଵTeభ×eభ=eమ՜ଵ,ଶ=ଶെTDiscount¾DiscountzThediscountd(t),foraterm(t)years,givesthepresentoneunitcurrency($1)bereceivedattheendthatterm.rtdt=1+dt=m0.5.51122.53Spot,ForwardRates¾¾¾ForwardRateandSpotRate0110.94%F଴.ହ,ଵ21.37%21+1+=1+2F଴.ହ,ଵ=1.79%7BasicMethodHowdeterminethepriceabond?ଵCଶC୘(1+S୘)C୲(1+y)P=++ڮ+=෍1+ଵ(1+Sଶ)ଶzSupposethata2-yearbondwithaprincipal$100providescouponsattherate6%perannumTheratescomputethebondprice:TreasuryZeroRatesMaturity(Years)ZeroRate(%)(continuouslyCompounded)0.55.011.5.02P=3eି଴.଴ହ×଴.ହ+3eି଴.଴ହ଼×ଵ+3eି଴.଴଺ସ×ଵ.ହ+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=0.9845BasicMethod¾¾¾PricingBondusingDiscountFactors,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,suchastaxtreatmentandReplicationMethod¾ExampleThreebondyieldsandpricesshownbelowꢀMaturityYTMCouponPrice(%par)95.2381231year2years2years5%6%6%0%0%6%99.00100The2-yearspotrateis6.2%.Isthereanarbitrageopportunityusingthesethreebonds?Ifso,describethetradesnecessaryexploitthearbitrageopportunity?ReplicationMethod¾Example$10001.06Bond1.06Bond2$100$6$6+$1001Bond30.06Bଵ+1.06Bଶ=Bଷ0.06×95.238+1.06×99.00=110.6543Bonds3isundervalued,sobuybond3.Bond1andBond2overvalued,thensellthemReplicationMethod¾ExampleTime=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)BasicMethod¾PriceanAnnuityzAnannuitywithsemiannualpaymentsisasecuritythatapaymentc/2everysixmonthsforTyearsbutneverafinal“principal”payment(i.e.,FV=0).ThepriceanA(T),isgivenby:Cy1A=1െy21+¾PriceanPerpetuityzAperpetuitybondisabondthatcouponsThepriceaperpetuityissimplythecoupondividedbytheyield.Priceaperpetuity=c/yzBondReturn¾GrossRealizedReturnsP+cെPR୲=PzzIfwantlookatthereturnoveralongerperiod,mustconsidertheinvestmentthecouponas.Supposethattheinitialpurchasepriceabondis$98,andthepurchaseoccurredimmediatelyafteracouponpaymentdate.Itearns$1.75couponeverysixmonths.Ayearlaterthepriceis98.7.Assumingthatthecouponcanbeinvestedatanannualizedrate2.2%(semi-annualcompounding),thegrossrealizedreturnis:98.7െ98+1.75+1.75×1+1.1%=0.043198BondReturn¾zzIncorporatesfundingcostP+cെBR୲=PContinuetheexample,assumingthatthefundsbuyfinancedat%perannum(semi-annualcompounding),thenetrealizedreturnis:8.7െ98×1+1.5%ଶ+1.75+1.75×1+1.1%39=0.012898BondReturnthecouponpaymentscanbereinvestedataninterestrateequaltotheyieldtomaturitythebondisheldmaturityIfthebondisnotheldtheinvestorfacestheriskthathemayhavesellforlessthanthepurchaseprice,resultingareturnthatislessthantheyieldknownasinterestraterisk.Reinvestmentriskexist.Futureinterestratescanbelessthantheyieldmaturityatthetimebondispurchased,knownasreinvestmentrisk.BondReturn¾YieldMaturityzTheYTMabondisthesinglediscountrateatwhichallcashflowsthebonddiscountedandsummeduptheprice.Example:Supposeabond$40everysixmonthsforfouryearsandafinalpayment$1,000atmaturityinfouryears.Ifthepriceis$850,calculatetheYTM.z9Answer:TheYTMistheythatsolvesthefollowingequation:$40y2$40y2$40+$1000$850=++ڮ+y21+1+1+FV=$1000;PV=-850;PMT=40;N=8ꢀꢁ&37ମ,ꢂ<6.46ମ<7012.92%BondReturn¾ExampleAbondwitha$100a4%couponannuallyfor3years.Thespotratescorrespondingthepaymentdatesasfollows:year1:4.5%;year2:5%;year3:5.5%.CalculatethepricethebondusingspotratesanddetermineintheYTMforthebond.zAnswer:Thepricethebondusingthespotratesisasfollows.44104P=++=96.02371.0451.05ଶ1.055ଷzComputetheYTM:441041+YTM96.0237=++1+YTM1+YTMFV=$100;PV=-ꢃꢄꢅꢆꢇꢈꢉꢀꢁ307ꢊꢀꢁ1ꢈꢀꢁ&37ମ,ꢂ<ꢋꢅꢊꢉମ<70ꢋꢅꢊꢉꢌBondReturn¾RelationshipbetweenSpotRatesandYTMଵ+YTMCFଶ1+YTMCF୬1+YTMP=++ڮ+1ଵ+RଵCFଶ1+RଶCF୬1+R୬=++ڮ+1zzYTMisakindaverageallthespotrates.Whenisaflattermstructurespotrates,theyieldmustequalthosespotrates.zzWhenthetermstructurespotratesisupward-sloping,theyieldthetwo-yearbondwillbebelowthetwo-yearspotrate(usetwo-yearbondasexample).Whenthetermstructurespotratesisdownward-sloping,theyieldthetwo-yearbondwillbeabovethetwo-yearspotrate.BondReturnInterestRateInterestRate98765439876543ForwardCurveSpotCurveYieldCurveYieldCurveSpotCurveForwardCurve012345678910012345678910Upward-SlopingDownward-SlopingBondReturn¾CouponEffectzThefactthatcorrectlypricedbondswiththesamematuritybutdifferentcouponshavedifferentyieldstomaturityiscalledthecouponeffect.zAsthecouponrises,theaveragetimeittakesbondholderstorecovertheircashflowsfalls.Therefore,thespotratesfortheearlypaymentdatesisbecomingmoreimportantindeterminingtheyieldtomaturity.领取考前题唯一微：xuebajun888s99upward-slopingtrend,theyieldmaturityfallsasthecouponrises.downward-slopingtrend,theyieldmaturityrisesasthecouponrises.BondReturn¾¾¾azThepriceasecurityisbydiscountingacashflowsusinganappropriatetermstructureplusaspread.ccP=++ڮ1+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:ķCarry-roll-down:thepricechangeduethepassagetimeratesmoveasexpectedbutwithnochangeinthespread.‹Themostcommonassumptionwhenthecarryroll-downiscalculatedisthatforwardratesrealized(i.e.,theforwardrateforafutureperiodremainsunchangedasmovethroughtime).P&LComponentsDecompositionP&LĸRatechange:thepriceeffectrateschangingtheintermediatetermstructuretothetermstructurethatactuallyattimet+1.ĹThepriceappreciationduetoaspreadchangeisthepriceeffectduetotheindividualspreadchangings(t)tos(t+1).P&LComponents¾ExampleStartperiod201020112012priceP&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%1101Pୡୟ୰୰୷ି୰୭୪୪ିୢ୭୵୬==94.3485=96.181+3.5%1+3.5%1+4.5%1101P୰ୟ୲ୣୡ୦ୟ୬୥ୣ=+1+2.5%11+2.5%1+3.5%101Pୱ୮୰ୣୟୢୡ୦ୟ୬୥ୣ=+=95.25771+3%1+3%1+4%Exercise1¾Thepriceofathree-yearzero-coupongovernmentbondis85.16.Thepriceofasimilarfour-yearbondis79.81.Whatistheone-yearimpliedforwardratefromyear3toyear4?A.5.4%B.5.5%C.5.8%D.6.7%¾¾Answer:D领取考前：xuebajun888s题唯一微信Exercise2¾Thefollowingdiscountfunctioncontainssemi-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%¾Answer:CExercise3¾Anannuity$10everyyearfor100yearsandcurrentlycosts$100.TheYTMisclosestto:A.5%B.7%C.9%D.10%¾CorrectAnswer:DExercise4¾A$1,000bondcarriesacouponrate10%,couponsandhas13yearsremainingtoMarketratescurrently9.25%.Thepricethebondisclosestto:A.$586.60B.$1,036.03C.$1,055.41D.$1,056.05¾CorrectAnswer:DExercise5¾Abondportfoliomanagerinvests$20millioninabondissuedatparthatmaturesin30years,andwhichpromisestopayanannualinterestrateof9%.Theinterestispaidonceperandthepaymentsarereinvestedatanannualinterestrateof8%.Thefirstpaymentisoneyearfromtoday.Whatistheannualyieldonthisinvestment?A.8.185%B.8.285%C.8.385%D.8.415%¾¾Answer:C领取考前：xuebajun888s题唯一微信Bond12..ShiftsNon-ParallelShiftsParallelStructureShifts¾¾One-FactorAssumptionzTheonlysignificantassumptionmadeabouthowthetermstructurechangesisthatallratechangesdrivenbyonefactoThesimplestone-factormodelassumesthatallratemovebythesameamount.DurationAnalysiszYielddurationmeasuresthesensitivityapricetoitsyieldchange.9Plainbond:ķMacaulayDuration&ModifiedDuration&DollarDuration&DV01ĸConvexity9Embeddedoption:ķEffectiveduration&EffectiveconvexityParallelStructureShifts¾¾MacaulayDurationzperiodcashflowreturningweightedbydiscountedcashPVCF୲PMac.D=෍×t=෍(w୲×t)ExampleIfabondhasapresentUSD93.06withacashflowinoneyearprovidingapresentUSD5.45andacashflowintwoyearsprovidingapresentUSD87.60,theMacaulaydurationwouldbe:5.4587.60×1+×2=1.941493.0693.06zzForaplainbond,theMacaulaydurationislessthanorequaltoitsForazerocouponbond,theMacaulaydurationequalstoitsParallelStructureShifts¾ModifiedDurationandDollarDurationz==1+z=×P×Pzz=.=×P=yParallelStructureShifts¾DV01zTheDV01istheabsolutevaluethepricechangeabondonepointchangeinyield.ȟꢀDV01=െȟ”DV01=MD×BondValue×0.0001¾Examplezz9Macaulayduration=1.9414Bondvalue=93.06Thisindicatesthata1bpschangeinthecontinuouslycompoundedyieldtogiverisetoapricechangeof:െ1.9414×93.06×0.0001=െ0.01807ParallelStructureShifts¾z9aaAAaBBF×F=aaaaz1===zzParallelStructureShifts¾Price-YieldrelationshipParallelStructureShifts¾¾LimitationsDurationzDurationprovidesagoodapproximationwhensmallparallelshiftintheinterestratetermstructure.itwillprovideapoorapproximationifnon-parallelshiftorthechangeis.ConvexityzAmeasurethenon-linearrelationship.Convexity=෍(w୲×tଶ)zWhenratesexpressedwithcompoundingmtimesperyear:ConvexityModifiedConvexity=y1+mRiskMetrics¾¾¾PriceApproximation1Py଴+ȟy=Py଴+fᇱy଴ȟy+fᇱᇱy଴ȟyଶ+ڮ2DollarDollarDurationConvexityzzzTheactual,exactprice:P=f(y଴+ȟy)Thedurationestimate:P=PെD×P×ȟyThedurationandconvexityestimate:1P=PെD×P×ȟy+×C×P×ȟy2ParallelStructureShiftsDurationandConvexityAnalysiszTheeffectparallelshiftsinterestratetermstructurecanbeaccuratebyaddingconvexityanalysistheanalysisduration.zIfCistheapproximatepricechangecanberefinedto:1ȟꢀ=െꢁꢀȟ›+CPȟ›2zThisapproximationallowsrelativelyparallelshiftstobeconsidered.ParallelStructureShiftsExampleConsidera10-yearzero-couponbondtradingatayield6%,semiannuallycompounding.Thepresentthebond:P=100/1+6%/2ଶ଴=55.368CalculationDurations,forazero-couponbond:zzzMacaulayduration=10yearsModifiedduration:MD=10/1+6%/2=9.708Dollarduration:DD=MD×P=9.708×55.368=537.553ParallelStructureShifts¾¾¾Example˄1˅canusethedurationtocalculatethebondpriceapproximatelywhentheyieldchanges,forexample,yieldgoesto7%6%TheexactpricebondisP=100/1+7%/2ଶ଴=50.257UsingdurationtoapproximateP=଴+fᇱy଴ȟ›=55.368െ537.553×1%=$49.993CalculateconvexityConvexity=141××21×20=98.9736%1+2Usingdurationandconvexitytoapproximate1P=55.368െ537.553×1%+×98.973×55.368×1%ଶ=50.2672ParallelStructureShiftsExample˄2˅Ifyieldgoes5%6%TheexactpricebondisP=100/1+5%/2ଶ଴=61.027UsingdurationtoapproximateP=଴+fᇱy଴ȟ›=55.368െ537.553×˄െ1%˅=$60.744Usingdurationandconvexitytoapproximate1P=55.368െ537.553×െ1%+×98.973×55.368×െ1%2=61.017ParallelStructureShiftsNegativeConvexityzAcallablebondgivestheissuertherighttoallorpartthebondbeforethespecifiedmaturitydate.zMostmortgagebondsnegativelyconvex,andcallablebondsusuallyexhibitnegativeconvexityatloweryields.1ParallelStructureShifts¾EffectiveDurationandEffectiveConvexityzInabond(withoutembeddedoptions)cantypicallyusemodifiedandeffectiveWhenthebondcontainsembeddedoptions,prefereffectiveduration:ȟꢀ/PPെPD=െ=ȟ”଴ȟ›999zP=initialobservedbondpriceȟ›=changeinrequiredyield(indecimalform)Effectiveconvexity:anapproximatemeasureconvexityDିെDାȟ›P+ାെ2P଴ȟyଶC==ParallelStructureShifts¾EffectiveDurationandEffectiveConvexityzExample:Supposethereisa10-yearbondwithanannualcouponof6%tradingatComputethebond’sdurationfora40basispointincreaseanddecreaseinyield,andcomputetheconvexityofthisbond.z9Answer领取考前：xuebajun888s题唯一微信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=െ103103െ97.11D=C==7.36252×100×0.004103+97.11െ2×100=68.75100×0.004ଶParallelStructureShifts¾HedgingbasedonEffectiveDurationz9999zSupposeTheeffectivedurationaninvestmentis୴ThetheinvestmentisVThedurationabondis୔ThethebondisPIfhedgeagainstsmallparallelshift,thenthepositioninthebondis:VD୚D୔P=ParallelStructureShifts¾¾¾HedgingbasedonEffectiveDurationandConvexityzcanusetwobondsbotheffectivedurationandeffectiveconvexityzero.Suppose999P,D,andCthevalue,duration,andconvexitythefirstbondP,D,andCthevalue,duration,andconvexitythesecondbondD୚,andC୚thevalue,duration,andconvexitythepositionthatistobehedged.zcanbothdurationandconvexitybychoosingଵandଶ:െVD୚െଵଵെଶDଶ=0VC୚+ଵଵ+ଶCଶ=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୅+େ=100,000,000େ୅00,000,000×4.7060+×14.9120=8.17551100,000,000୅=66m,େ=34mC୮୭୰୲୤୭୪୧୭=0.66×25.16+0.34×331.73=129.4֜Exercise1¾Abondhasacouponrateof6%perannum(thecouponsarepaidsemiannually)andasemiannuallycompoundedyieldof4%perannum.Thebondmaturesin18monthsandthenextcouponwillbepaid6monthsfromnow.Whichnumberbelowisclosesttothebond’sMacaulayduration?领取考前题唯一微：xuebajun888sA.1.023yearsB.1.457yearsC.1.500yearsD.2.915years¾CorrectAnswer:BExercise2¾Azero-couponbondwithamaturity10yearshasanannualeffectiveyield10%.Whatistheclosestforitsmodifiedduration?A.9B.10C.99D.100¾Answer:AExercise3¾Themodifieddurationis10.46abondwithacurrentprice$716.38.WhatistheDV01?A.$0.40B.$0.75C.$1.25D.Needinformation(yield,maturity)¾Answer:BExercise4¾Abondportfoliohasthefollowingcompositions: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:AExercise5¾A8%semiannualcouponbondwith$100currentlytradesat$78.75andhasaneffectiveduration9.8yearsandaconvexity130.Whatisthepricethebondiftheyieldfallsby150basispoints?A.$67.17B.$86.47C.$91.48D.$95.43¾CorrectAnswer:CExercise6¾Whichofthefollowingstatementsisalwayscorrect?A.Nomethodcanhedgeinterestraterisk.B.Single-factormodelassumemean-reversionbetweenoneshort-termandonelong-termrate.C.Single-factormodelsassumerisk-freesecuritieshavecreditexposure.D.Single-factormodelsassumethatallratechangesaredrivenbyone领取考前押题唯一微：xuebajun888s¾Answer:DNon-ParallelStructureShifts¾ModelingNon-ParallelStructureShiftszInpractice,theremanydifferenttypesnon-parallelshifts.Sometimesshort-termratesmovedownwhilelong-termmoveup,orviceversa.Occasionally,shortlong-terminterestratesmoveinonedirection,whilemedium-termratesmoveintheotherdirectionyNonparallelTNon-ParallelStructureShifts¾PrincipalComponentsAnalysiszAstatisticaltechniquecalledprincipalcomponentsanalysiscanbeusedtounderstandtermstructurechangesinhistoricaldata.Principalcomponentanalysiscananalyzetheeffectsofmultiplefactorsandestimatetheirrelativeimportanceindescribingmovementsinthetermstructure.zzThistechniquelooksatthedailychangesininterestratescorrespondingtovariousmaturitiesandidentifiesfactorsthathavethefollowingcharacteristics:领取考前题唯一微：xuebajun888sķThesefactorsuncorrelated;ĸDailychangesintermstructurelinearcombinationsthefactors;ĹThefirsttwoorthreefactorsaccountforthemajoritytheobserveddailymovements.Non-ParallelStructureShifts¾෍௜݂zz୧Non-ParallelStructureShifts¾Forourdata,thetotalvarianceis14.15ଶ+4.91ଶ+ڮ+0.68ଶ=235.77zzThefirstfactoraccountsfor84.9%(=14.15ଶ/235.77)Thefirsttwofactorsaccountfor:14.15ଶ+4.91ଶ=95.14%235.77zThefirstthreefactorsaccountfor97.66%thevariance.14.15ଶ+4.91ଶ+2.44ଶ=97.66%235.77Non-ParallelStructureShifts¾Key-RateExposurez9Key-RateShiftsAssumptionThecrucialassumptiontherateisthatallratescanbedeterminedasafunctionofarelativelysmallnumberofkeyrates.Afewratesalongtheterm-structurearepickedwhicharerepresentativeofthecurve.领取考前题唯一微：xuebajun888s99Therateagivenmaturityisaffectedsolelybyitsclosestkey-rate.Shiftsinthekey-ratesdeclinelinearly.1056Non-ParallelStructureShifts¾Key-RateExposurez99Key-RateShiftsMetricsRate’01s:whichistherateequivalentDV01.RateDuration:whichistherateequivalentdurations.ȟPȟy1PȟPȟyDV01୩ୣ୷=െ0.0001×D୩ୣ୷=െڄzExample:Calculate30-yearrate01andratedurationapplyingaonebasispointshift.=0.1033=41.129425.01254െ25.11584KeyRate01ଷ଴=െ0.0001×0.01%5.01254െ25.115842KeyRateDurationଷ଴=െ25.11584×0.01%Non-ParallelStructureShifts¾¾¾Example1SupposeaportfolioconsistsaUSD1millioninvestmentineachaone-and15-yearcouponbond.Thedecreaseintheforaone-basis-pointincreaseinthespotratesisasfollows:Decreaseinfora1bpsIncreaseinSpotRatesSpotRateMaturity135915PortfolioDecrease96.72270.26433.85677.93957.66Supposefive-yearandten-yearinterestrateselectedasratetobeusedinanalysis,then:6Non-ParallelStructureShiftsExample1ChangesinSpotRateforChangesinRateSpotRateMaturityRateShift1100350109150Five-year0.66670.3333000.20.801KR01sforPortfolioCalculationPartial01ଵ276.9096.72+(0.6667×270.26)KR01ଶ0.3333×270.26+433.85+(0.2×677.93)659.520.8×677.93+957.661,500.00KR01ଷNon-ParallelStructureShiftsExample2Supposethreedifferenthedginginstrumentsusedtohedgerisks.TheKR01sportfolioandhedginginstrumentsisshowninthefollowingtable:DataforHedgingUsingKR01sPortfolioHedgingInstruments1402636KR011KR012KR0132524767704448825024752+40xଵ+6xଶ+6xଷ=076+4xଵ+44xଶ+8xଷ=070+2xଵ+8xଶ+50xଷ=0xଵ=െ3,xଶ=െ8,xଷ=െ14Non-ParallelStructureShifts¾ForwardBucketShiftzEachforward01iscomputedbyshiftingtheforwardratesinthatbucketbyonebasispoint.Non-ParallelStructureShifts¾EstimatingPortfoliozzzRegulatorsBankstoconsidertendifferentKR01swhenanalyzingtherisktheirportfolios(including3-month,6-month,and30-yearspotrates).BanksneedalltheKR01szero,buttheyneeduseKR01exposureandstandarddeviationthetenratestoestimateriskmeasuressuchasorExpectedShortfall.Theformulaforthestandarddeviationthechangeintheportfolioinonedayis:ɐ୔=෍෍ɏ୧୨ɐ୧ɐ୨×KR01୧×KR01୨ɐ୧isthevolatilitythedailymovementinratei(measuredinbps)andɏ୧୨isthecorrelationbetweenthedailymovementsinrateiandj.Exercise¾Assumeyouownasecuritywitha2-yearrateexposure$4.78,andyouwouldtohedgeyourpositionwithasecuritythathasacorresponding2-yearrateexposure0.67per$100facevalue.Whatamountfacewouldbeusedhedgethe2-yearexposure?A.$478B.$239C.$713D.$670¾Answer:COption1.2.3.OtherAssetsOne-Step¾Risk-Neutralzdefinerisk-neutralworldasoneinvestorsdonotadjusttheirexpectedreturnbasedrisk,sotheexpectedreturnallassetsisrisk-freeinterestrate.zTherisk-neutralvaluationprinciplestatesthatifassumeinarisk-neutralworld,cangetafairpriceforaderivative.One-Step¾ExampleThestockpriceABCcompanyis$20Thestockwillgoup2ordown18threemonthsWhattheEuropeancallpricez2thisstockmonthsnow?SupposethestrikepriceisK=$21,continuouslycompoundedrisk-freerateis12%.Price=$22pOptionPrice=$1Price=$20OptionPricef1–pPrice=$18OptionPrice=$0One-Step¾Solution1zzzzzLong©stock,short1call.Inarisk-neutralworld:22©–1=18©©=0.25att0=20©–fatt1=22©–120×0.25െfe଴.ଵଶ×଴.ଶହ=22×0.25െ1zf=0.633¾Solution222P+181െP=20eP=0.6523f=1×p+0×1െpeି଴.ଵଶ×଴.ଶହ=0.633One-Step¾Generalizationupu=u–K,=0f1–pdd=d–K,zzLong©stocksandshort1calloptionhedgetherisk.S଴—ȟെf୳=S†ȟെf,eliminaterisksinbothcases.fെfୢS଴uെS଴dȟ=f=pf୳+1െpfୢeeെduെdp=u=ed=eMulti-Step¾EuropeanOptionsS0uufS0SfS0udfudfS0ddfeെduെdp=f=epଶf୳୳+2p1െpf୳ୢ+1െpଶfୢୢMulti-Step¾¾¾Example:EuropeanCallOptionwithUpandDownInformedbyAssetTimeRisklessYield5%2%$810$8000.520%udp1.10520.90480.5126$989.34$895.19732.92$810$810$$663.17Multi-StepExample:EuropeanCallOptionwithUpandDownInformedbyu=e=eଶ଴%×଴.ଶହ=1.1052d=0.9048e(୰ି୯)୼୲െduെdeെ0.90481.1052െ0.9048p===0.5126189.34100.66.0653.41050(189.34×0.5126+10×1െ0.5126)݁ି଴.଴ହ×଴.ଶହ=100.6610×0.5126+0×1െ0.5126݁ି଴.଴ହ×଴.ଶହ=5.06100.66×0.5126+5.06×1െ0.5126݁ି଴.଴ହ×଴.ଶହ=53.4Multi-StepAmericanOptions0ztheoptionateachnodeisnolessthantheintrinsicvalue.Sf0udf0Multi-Step¾Example:AmericanPutOptionwithpricejump+/-20%AssetTime50$52RisklessYield5%0%$2udp282$72$6040$4832$50$$Multi-Step¾Example:AmericanPutOptionwithpricejump+/-20%01.4147/05.09/249.4634/12200×0.6282+4×1െ0.6282eି଴.଴ହ×ଵ=1.4147×0.6282+20×1െ0.6282eି଴.଴ହ×ଵ=9.4634.4147×0.6282+12×1െ0.6282eି଴.଴ହ×ଵ=5.0941OtherAssets¾¾OptionsonwithDividendseെdp=Everythingelseabouttheisthesameasbefore.OptionsonIndicesuെdzzindexprovidesadividendyield.thevaluationshouldinvolvethemodificationasabove.¾¾OptionsonCurrenciesCurrencycanbeconsideredasanassetprovidingayield.OptionsonFutureszzItcostsnotingintoafuturescontractandcanacontractastockpayingadividendyieldTherefore,get:1െdp=uെdMulti-Step¾IncreasingtheNumberTimePeriodszAsthenumbertimestepsisincreased(sothat଱Wbecomessmaller),thebinomialmodelmakesthesameassumptionsaboutstockpricebehaviorastheBlack-Scholes-Mertonmodel.WhenthebinomialisusedpriceaEuropeanoption,thepriceconvergestheBlack-Scholes-Mertonprice,asexpected,asthenumbertimestepsisincreased.Exercise1¾Astockcurrentlytradesat$10.theendthreemonths,thestockwilleitherbe$11or$9.Thecontinuouslycompoundedrisk-freerateinterestis3.5%perThea3-monthEuropeancalloptionwithastrikeprice$10isclosestto:A.$0.11B.$0.54C.$0.65D.$1.01¾Answer:BExercise2¾AtraderhasanAmericanputoptionwithstrikeprice$50.Theunderlyingassetisstockwithaspotprice$40.Usinganone-stepbinomialtoevaluatetheoption.Supposethestockpricewillgoupordownby$8in6month,therisk-freerateis6.2%,whatisthethisAmericanA.USD8.19B.USD8.45C.USD10.00D.USD10.32¾Answer:COptionModelAssumptions¾AssumptionszzThestockpricefollowstheprocesswithnjandIJconstant.notransactioncostsortaxes.Allsecuritiesperfectlydivisible.zzzzzTherenodividendsduringthelifethederivative.Therenorisklessarbitrageopportunities.Securitytradingiscontinuous.Therisk-freerateinterest,isconstantandthesameforallmaturities.Theoptionsbeingconsideredcannotbeexercised.PriceMovement¾LognormalAssumptionzzGeometricBrownianmotiondS୲=uSdt+ɐSdZLemma߲f߲f12߲ଶf߲fdft,S=+uS୲+ɐଶSଶdt+ɐS୲dZ୲߲t߲S߲Sଶ߲SzzAsthestockpricefollowsaGeometricBrownianMotion,ifdefineG=lnS,usinglemma,have:1dG=uെɐଶdt+ɐdZ2Avariablehasalognormaldistributionifthenaturallogarithmthevariableisnormallydistributed.WhiletheequationaboveshowsthatlnS୘isnormallydistributed.Thestockpriceislognormallydistributed.PricingFormulas¾¾B-SdifferentialequationV=eEmaxS୘െK,0ThePriceEuropeanOptionscall=S଴NଵെKeNdଶput=KeNെdଶെS଴Nെଵɐଶ2lnS଴/K+r±Tଵ,ଶ=ɐTzzzNଵisthedeltaofcallNdଶistheprobabilityofcallexercise1െNdଶistheprobabilityofputexercisePricingFormulas¾ExamplezzzzzEuropeancalloptionpriceis$10,is$9.is20%.issixmonths.Risklessrateis5%.20%ଶ2ln10/9ln10/9+25%+×0.5×0.5ଵ==0.992=0.8510%×0.50%2+5%െ2dଶ=20%×0.5call=10N0.992െ9eି଴.଴ହ×଴.ହN0.851=1.35OtherAssets¾OptionsonwithDividendsc=S଴eNଵെKeNdଶp=KeNെdଶെS଴eNെଵlnS଴/K+rെq±ɐଶ/2Tଵ,ଶ=ɐT¾¾OptionsonCurrencieszBehavesastockpayingadividendyieldattheforeignrisk-freerate.OptionsonFutureszBehavesastockpayingadividendyie