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TermStructureModels
Black’smodelisconcernedwithdescribingtheprobabilitydistributionofasinglevariableatasinglepointintime
Atermstructuremodeldescribestheevolutionofthewholeyieldcurve
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TheZeroCurve
Theprocessfortheinstantaneousshortrate,r,inthetraditionalrisk-neutralworlddefinestheprocessforthewholezerocurveinthisworld
IfP(t,T)isthepriceattimetofazero-couponbondmaturingattimeT
whereistheaveragerbetweentimestandT
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EquilibriumModels(RiskNeutralWorld)
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MeanReversion(,page684)
Interest
rate
HIGHinterestratehasnegativetrend
LOWinterestratehaspositivetrend
Reversion
Level
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AlternativeTermStructuresinVasicek&CIR(,page686)
ZeroRate
Maturity
ZeroRate
Maturity
ZeroRate
Maturity
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PropertiesofVasicekandCIR
P(t,T)=A(t,T)e−B(t,T)r
TheAandBfunctionsaredifferentforthetwomodels
Thesecanbeusedtoprovidealternativedurationandconvexitymeasures
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BondPriceProcessesinaRiskNeutralWorld
FromIto’slemma
Marketpriceofinterestrateriskappearstobeabout−
Thiscanbeusedtoconvertarealworldprocesstoarisk-neutralprocessorviceversa
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EquilibriumvsNo-ArbitrageModels
Inanequilibriummodeltoday’stermstructureisanoutput
Inano-arbitragemodeltoday’stermstructureisaninput
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DevelopingNo-Arbitrage Modelforr
Amodelforrcanbemadetofittheinitialtermstructurebyincludingafunctionoftimeinthedrift
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