期权、期货及其他衍生产品课件4金融工程学

4.1InterestRatesChapter44.2TypesofRatesTreasuryratesLIBORrates(LIBIDrates)(LIBORLIBID)Reporates色诺芬（CCER）金融研究数据库4.3MeasuringInterestRatesThecompoundingfrequencyusedforaninterestrateistheunitofmeasurementThedifferencebetweenquarterlyandannualcompoundingisanalogoustothedifferencebetweenmilesandkilometers4.4SupposethatanamountAisinvestedfornyearsataninterestrateofRperannum.Iftherateiscompoundedonceperannum,theterminalvalueoftheinvestmentisIftherateiscompoundedmtimesperannum,theterminalvalueoftheinvestmentisWhenm=1,therateissometimesreferredtoastheequivalentannualinterestrate.4.54.6ContinuousCompounding(Page79)Inthelimitaswecompoundmoreandmorefrequentlyweobtaincontinuouslycompoundedinterestrates$100growsto$100eRTwheninvestedatacontinuouslycompoundedrateRfortimeT$100receivedattimeTdiscountsto$100e-RTattimezerowhenthecontinuouslycompoundeddiscountrateisR4.7ConversionFormulas(Page79)DefineRc:continuouslycompoundedrateRm:equivalentratewithcompoundingmtimesperyearRmRmRmecmmRmcln/114.8ZeroRatesAzerorate(orspotrate),formaturityTistherateofinterestearnedonaninvestmentthatprovidesapayoffonlyattimeT4.9Example(Table4.2,page81)Maturity(years)ZeroRate(%contcomp)0.55.01.05.81.56.42.06.84.10BondPricingTocalculatethecashpriceofabondwediscounteachcashflowattheappropriatezerorateInourexample,thetheoreticalpriceofatwo-yearbondprovidinga6%perannumsemiannuallyis333103983900505005810006415006820eeee.........4.11BondYieldThebondyieldisthediscountratethatmakesthepresentvalueofthecashflowsonthebondequaltothemarketpriceofthebondSupposethatthemarketpriceofthebondinourexampleequalsitstheoreticalpriceof98.39Thebondyield(continuouslycompounded)isgivenbysolvingtogety=0.0676or6.76%.39.981033330.25.10.15.0yyyyeeee4.12ParYieldTheparyieldforacertainmaturityisthecouponratethatcausesthebondpricetoequalitsfacevalue.Inourexamplewesolveg)compoundins.a.(withgetto87610021002220.2068.05.1064.00.1058.05.005.0.c=ecececec4.13ParYieldcontinuedIngeneralifmisthenumberofcouponpaymentsperyear,disthepresentvalueof$1receivedatmaturityandAisthepresentvalueofanannuityof$1oneachcoupondateAmdc)100100(4.14SampleData(Table4.3,page82)BondTimetoAnnualBondCashPrincipalMaturityCouponPrice(dollars)(years)(dollars)(dollars)1000.25097.51000.50094.91001.00090.01001.50896.01002.0012101.64.15TheBootstrapMethodAnamount2.5canbeearnedon97.5during3months.The3-monthrateis4times2.5/97.5or10.256%withquarterlycompoundingThisis10.127%withcontinuouscompoundingSimilarlythe6monthand1yearratesare10.469%and10.536%withcontinuouscompounding4.16TheBootstrapMethodcontinuedTocalculatethe1.5yearratewesolvetogetR=0.10681or10.681%Similarlythetwo-yearrateis10.808%96104445.10.110536.05.010469.0Reee4.17ZeroCurveCalculatedfromtheData(Figure4.1,page84)910111200.511.522.5ZeroRate(%)Maturity(yrs)10.12710.46910.53610.68110.8084.18ForwardRatesTheforwardrateisthefuturezerorateimpliedbytoday’stermstructureofinterestrates4.19CalculationofForwardRatesTable4.5,page85ZeroRateforForwardRateann-yearInvestmentfornthYearYear(n)(%perannum)(%perannum)13.024.05.034.65.845.06.255.36.50.0310.0510.042100100eee0.0420.05810.0463100100eee4.20FormulaforForwardRatesSupposethatthezeroratesfortimeperiodsT1andT2areR1andR2withbothratescontinuouslycompounded.TheforwardratefortheperiodbetweentimesT1andT2is4.21InstantaneousForwardRateTheinstantaneousforwardrateforamaturityTistheforwardratethatappliesforaveryshorttimeperiodstartingatT.ItiswhereRistheT-yearrateTRTR122121()FTRRRRTT4.22ForwardRateAgreementAforwardrateagreement(FRA)isanagreementthatacertainratewillapplytoacertainprincipalduringacertainfuturetimeperiod4.23ForwardRateAgreementcontinuedAnFRAisequivalenttoanagreementwhereinterestatapredeterminedrate,RKisexchangedforinterestatthemarketrateAnFRAcanbevaluedbyassumingthattheforwardinterestrateiscertaintoberealized4.244.254.26ValuationFormulas(equations4.9and4.10page88)ValueofFRAwhereafixedrateRKwillbereceivedonaprincipalLbetweentimesT1andT2isValueofFRAwhereafixedrateispaidisRFistheforwardratefortheperiodandR2isthezerorateformaturityT2WhatcompoundingfrequenciesareusedintheseformulasforRK,RM,andR2?22))((12TRFKeTTRRL22))((12TRKFeTTRRL4.27DurationofabondthatprovidescashflowciattimetiiswhereBisitspriceandyisitsyield(continuouslycompounded)Thisleadsto1iytniiiceDtByDBBDuration(page89)4.28DurationContinuedWhentheyieldyisexpressedwithcompoundingmtimesperyearTheexpressionisreferredtoasthe“modifiedduration”myyBDB1Dym14.29ConvexityTheconvexityofabondisdefinedas221221sothat1()2inytiiicteBCByBBDyCyB4.304.31TheoriesoftheTermStructurePage93ExpectationsTheory:forwardratesequalexpectedfuturezeroratesMarketSegmentation:short,mediumandlongratesdeterminedindependentlyofeachotherLiquidityPreferenceTheory:forwardrateshigherthanexpectedfuturezerorates4.28证券组合A由一个账面价值为2000的1年期零息债券和一个账面价值6000的10年期零息债券组成。债券组合B由一个账面价值为5000的5.95年期的零息债券组成。当前所有债券的收益率为10%p.a.请证明两个债券组合的久期相同。4.32账面价值为2000的债券现值为：账面价值为6000的