Market price of risk specifications for affine models

48 Pages·2006·0.35 MB·English

ARTICLE IN PRESS
JournalofFinancialEconomics83(2007)123–170
www.elsevier.com/locate/jfec
Market price of risk specifications for affine models:
$
Theory and evidence
Patrick Cheriditoa, Damir Filipovic´b, Robert L. Kimmelc,(cid:2)
aPrincetonUniversity,DepartmentofOperationsResearchandFinancialEngineering,Princeton,NJ08544,USA
bUniversityofMunich,DepartmentofMathematics,80333Munich,Germany
cOhioStateUniversity,FisherCollegeofBusiness,Columbus,OH,43210,USA
Received14December2004;receivedinrevisedform15September2005;accepted23September2005
Availableonline7September2006
Abstract
Weextendthestandardspecificationofthemarketpriceofriskforaffineyieldmodels,andapply
it to U.S. Treasury data. Our specification often provides better fit, sometimes with very high
statistical significance. The improved fit comes from the time-series rather than cross-sectional
features of the yield curve. We derive conditions under which our specification does not admit
arbitrage opportunities. The extension has extremely strong statistical significance for affine yield
models with multiple square-root type variables. Although we focus on affine yield models, our
specification canbeused withotherasset pricingmodelsas well.
r2006Elsevier B.V.All rightsreserved.
JELclassifications:C51;G12;G13
Keywords:Termstructure;Marketpriceofrisk;Affineyieldmodels;No-arbitragepricing
$WewouldliketothankJunLiu,GeorgeChacko,ChrisJones,ananonymousreferee,seminarparticipantsat
the Canadian Mathematical Society, Princeton University, Northwestern University, the University of Illinois
Urbana-Champaign, Duke University, Ohio State University, the Triangle Econometrics Workshop, the
EconometricSociety,theCentreforAdvancedStudiesinFinanceattheUniversityofWaterloo,theUniversityof
CaliforniaSantaBarbara,theWesternFinanceAssociation,theBachelierSocietyThirdWorldCongress,andthe
ChineseAcademyofSciencesCentreforStatisticalResearchformanyhelpfulcommentsandsuggestions.Any
remainingerrorsaresolelyourresponsibility.WewouldalsoliketothankRobertBlissandGregDuffeeforthe
datasetusedinthisstudy.
(cid:2)
Correspondingauthor.Fax:[phone].
E-mailaddress:[email protected](R.L.Kimmel).
0304-405X/$-seefrontmatterr2006ElsevierB.V.Allrightsreserved.
doi:10.1016/j.jfineco.2005.09.008
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124 P.Cheriditoetal./JournalofFinancialEconomics83(2007)123–170
1. Introduction
The square-root process of Feller (1951) has been used widely in financial economics,
appearing in term structure models such as Cox et al. (1985) and stochastic volatility
models of equity prices such as Heston (1993). Multivariate extensions of the square-
root process have appeared in the term structure literature; see, for example, Duffie
and Kan (1996), Dai and Singleton (2000), and Duffie (2002). The widespread use of
this process is undoubtedly due at least in part to its relatively straightforward ana-
lytical properties: in the square-root process, a state variable follows a diffusion in
whichboththedriftandthediffusioncoefficientsareaffinefunctionsofthestatevariable
itself. Of course, a model for asset prices must specify not only the stochastic
process followed by a set of underlying factors, but also the attitude of investors towards
the risk of those factors. Since the pioneering work of Harrison and Kreps (1979) and
Harrison and Pliska (1981), this task is often accomplished by specifying the behavior of
the state variable(s) under both an objective probability measure and an equivalent
martingale measure. A common practice is to have the state variables follow a Feller
square-root process under both probability measures, but with different governing
parameters.
Thislatterobjectiveisnormallymetbyassigningtoeachstatevariableamarketpriceof
risk process that is proportional to the square root of that state variable. Since the
instantaneous volatility of each state variable is also proportional to its square root, the
productofthemarketpriceofriskandvolatilityisproportionaltothestatevariableitself.
Subtraction of this product from the drift under the objective probability measure
therefore resultsinadriftundertheequivalentmartingalemeasurethat isalso affine. Ifa
process iswithin theFeller square-rootclass underthe objective probability measure, this
market price of risk specification ensures that it is within the same class under the
equivalentmartingalemeasureaswell.Amarketpriceofriskthatisinverselyproportional
tothesquarerootofthestatevariablewouldalsoretaintheaffinityofthedriftunderboth
measures. However, such a market price of risk specification is rarely used in financial
modeling.1Coxetal.(1985)discussthispossibility,andpointoutthatitleadstoarbitrage
opportunitiesiftheboundaryvalueoftheprocesscanbeachieved:whiletheinstantaneous
volatilityofthestatevariableiszeroattheboundary,ifthemarketpriceofriskisinversely
proportionaltothesquarerootofthestatevariable,theriskpremiumassociatedwiththat
state variable does not go to zero as the volatility approaches zero. Ingersoll (1987)
imposes the condition that the risk premium goes to zero as volatility goes to zero in a
similar setting. Bates (1996) also imposes this condition in a stochastic volatility model;
ChernovandGhysels(2000),workinginasimilarsetting,discussthetypeofmarketprice
specification we propose, but leave unresolved the issue of whether it precludes arbitrage
opportunities.Inarecenttermstructureapplication,Duffee(2002)specificallyavoidsthis
market price of risk specification. However, whether or not a Feller square-root process
can achieve the boundary value depends on the values of the governing parameters. For
someparametervalues,theinstantaneousvolatilityofthestatevariablecanapproachzero
arbitrarily closely but never actually achieve this value. The market price of risk can then
1Inamodelforstochasticvolatilityofequityprices,Erakeretal.(2003)considerajump-diffusionmodelin
whichifthejumppartofthemodelisignored,themarketpriceofriskisofthisform.However,itisnotexplicitly
identifiedandtheabsenceofarbitrageisnotformallydemonstrated.
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be arbitrarily large, but finite, when the state variable takes values near zero. It is not
immediately clear whether arbitrage opportunities exist in this case; we show that they
do not.
Although the reason for the avoidance of this market price of risk specification in the
literature is not clear, it may be related to the difficulty of proving that it does not
offer arbitrage opportunities. Specifically, it is quite difficult or impossible to determine
whether this specification satisfies conventional criteria, e.g., those of Novikov or
Kazamaki. However, these criteria are sufficient but not necessary to prove that the
Girsanovratioisamartingale.UsingtheapproachofCheriditoetal.(2005),weshowthat
this market price of risk specification does not offer arbitrage opportunities, provided
certain parameter restrictions are observed. Using the extended market price of risk
specification, we estimate several affine term structure models (specifically, all nine
canonical families of affine models with one, two, or three factors, as Dai and Singleton,
2000,describe),andcomparetheresultstothosethatobtainusingmoretraditionalmarket
price of risk specifications. Although there are nine distinct canonical families, our
extension is degenerate (i.e., is the same as the specification of Duffee, 2002) for three of
thefamilies.Fortheremainingsixfamilies,wefindthattheextendedspecificationusually
resultsinastatisticallysignificantimprovementinthefitofaffineyieldmodelstodataon
U.S.Treasurysecurities.Theimprovementisparticularlystrongforthethreemodelswith
multiple semi-bounded state variables. To determine the cause of the statistical
improvement, we explore both the fit of the cross-sectional shape of the yield curve (i.e.,
thedifferencebetweentheshapeoftheyieldcurvepredictedbyanestimatedmodelandthe
shapeoftheyieldcurveobservedinthedata)andtheaccuracyofthepredictedtime-series
behavior. Our extended market price of risk specification appears to offer little
improvement in the cross-sectional fit of the yield curve, and actually results in a slight
degradationforsomemodels.However,thetime-seriesbehaviorpredictedbymodelsusing
the extended market price of risk specification is often substantially more accurate than
that predicted by more traditional specifications. This improvement almost always
manifests itself in reduced bias of yield forecasts, but for some models, the volatility of
yield changes is also modelled much more accurately. Among three-factor models, some
authors(e.g.,DaiandSingleton,2000,whointroducethemodelclassificationschemeand
notation we use here) find that one particular model, the A (3) model, captures many
1
features of term structure behavior more accurately than other three-factor models. With
theintroductionofourmarketpriceofrisk,thefitoftheA (3)modelimproves,butthefit
1
of two other three-factor models, namely, the A (3) and A (3) models, improves
2 3
substantially more. The relatively larger improvement in these latter two models could
possibly reverse the preference ordering of three-factor models once the market price of
risk is generalized.
The rest of this paper is organized as follows. In Section 2, we describe a class of
multivariate term structure models driven by square-root processes, and define the
admissible change of measure using our extended market price of risk specification. In
Section 3, we show that this specification precludes arbitrage opportunities. In Section 4,
we describe the data and estimation procedure we use to estimate and test our
specification. In Section 5, we present the results and show that the extended market
price of risk specification offers significantly better fit to the data than do standard
specifications for most models, especially those with two or more square-root-type state
variables. Finally, Section 6 concludes.
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126 P.Cheriditoetal./JournalofFinancialEconomics83(2007)123–170
2. Models
Throughout, we focus on affine yield models of the term structure of interest rates,
which we define as follows.
Definition 1. An affine yield model of the term structure of interest rates isa specification
of interest rate and bond price processes such that:
1. The instantaneous interest rate r is an affine function of an N-vector of state variables
t
denoted by Y,
t
r ¼d þdTY , (1)
t 0 t
whered isaconstantanddisanN-vector.Wesometimesrefertoindividualelements
0
of the vector y, using the notation y(k) for 1pkpN.
t t
2. The state variables Y follow the diffusion process:
t
dY ¼mPðY ÞdtþsðY ÞdWP, (2)
t t t t
where mP (Y) is an N-vector, s (Y) is an N(cid:2)N matrix, and WP is an N-dimensional
t t t
standard Brownian motion under the objective probability measure P.
3. Theinstantaneousdrift(underthemeasureP)ofeachstatevariableisanaffinefunction
of Y,
t
mPðY Þ¼aPþbPY , (3)
t t
for some N-vector aP and some N(cid:2)N matrix bP.
4. The instantaneous covariance between any pair of state variables is an affine function
of Y,
t
(cid:2)sðY ÞsTðY Þ(cid:3) ¼a þbTY , (4)
t t i;j ij ij t
where [s(Y)sT(Y)] denotes the element in row i and column j of the product
t t i,j
s(Y)sT(Y), a is a constant, and bT is an N-vector for each 1pi,jpN.
t t ij ij
5. There exists a probability measure Q, equivalent to P, such that Y is a diffusion
t
under Q:
dY ¼mQðY ÞdtþsðY ÞdWQ, (5)
t t t t
wheremQ(Y)isanN-vector,WQisanN-dimensionalstandardBrownianmotionunder
t t
Q, and the drift of each state variable is an affine function of the state vector
mQðY Þ¼aQþbQY (6)
t t
for some N-vector aQ and some N(cid:2)N matrix bQ.
6. Prices of zero-coupon bonds are conditional expectations of the discounted payoffs
under the measure Q:
(cid:4) RT (cid:5)
Bðt;TÞ¼EQt e(cid:3) t rudu . (7)
Feller (1951) treats existence of a process satisfying the second, third, and fourth
conditions in a univariate setting, and Duffie and Kan (1996) do so in a multivariate
setting. Duffie et al. (2003) provide a general mathematical characterization of affine
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processes, including those with jumps. The diffusions we consider here are special cases.
Existencecanessentiallybecharacterizedasarequirement thatthestatevectorY remain
t
withinaregioninwhichs(Y)sT(Y)ispositivesemidefinite.Moreformally,itsufficesthat
t t
there exist constants g ,y,g and nontrivial N-vectors h ,y,h such that s(Y)sT(Y) is
1 M 1 M t t
positive definite2 if and only if
g þhTY 40 (8)
i i t
foreachvalueof1pipM.Wedenotetheregioninwhichthisconditionissatisfied(forall
i)byD,andtheclosureofDbyD¯.Notethats(Y)sT(Y)ispositivedefiniteinD,positive
t t
semidefiniteinD¯,andnotpositivesemidefiniteoutsideD¯.Certainconditionsmustholdon
the boundaries of D to ensure that the state vector cannot leave the region D¯. For each
value of YAD¯, we must have:
t
(cid:6)g þhTY ¼0(cid:7))(cid:6)hTmPðY ÞX0(cid:7), (9)
i i t t
(cid:6)g þhTY ¼0(cid:7))(cid:6)hTsðY ÞsTðY Þh ¼0(cid:7) (10)
i i t t t i
for each value of i. Intuitively, these two requirements are (1) the drift must not pull the
statevectorY outoftheregionD¯,sinces(Y)sT(Y)thenfailstobepositivesemidefinite,
t t t
and(2)thevolatilitymustnotallowY tomovestochasticallyoutofD¯.Ofcourse,wemust
t
alsohaveY AD¯.Itispossiblethatm¼0,i.e.,thatDistheentirespaceRN,inwhichcase
0
therestrictionsofEqs.(9)and(10)areentirelyvacuous.Therearenoseparateuniqueness
criteria; if a solution to Eq. (2) exists, it is unique.3
Inadditiontoexistenceanduniqueness,achievementofboundaryvaluesisofparticular
importance when analyzing our market price of risk specification. Intuitively, within the
region D, the drift of the state vector must not only satisfy the existence condition of
Eq.(9),butalsopullthestatevectorbacktowardtheinteriorofDwithsufficientstrength
to ensure that the boundary cannot be achieved. Feller (1951) and Ikeda and Watanabe
(1981)treattheunivariatecase;DuffieandKan(1996)treatthemorecomplexmultivariate
case. However, possibly after changing the coordinate system, all the models we consider
in this paper are such that the region D is of the form (0,N)M(cid:2)RN(cid:3)M,M¼0,y,N, in
whichcaseitiseasytoderivesufficientboundarynonattainmentconditionsfromtheone-
dimensional case. We always impose boundary nonattainment conditions; we refer to the
first M state variables as restricted and the last N(cid:3)M as unrestricted.
Withrespecttopossiblechangesofthecoordinatesystem,notethatanytransformation
X ¼AþBY (11)
t t
for some N-vector A and some regular N(cid:2)N matrix B of a given affine yield model with
statevariablesY constitutesanotheraffineyieldmodelthatcanproduceexactlythesame
t
short-rate processes r as the riginal model. To ensure identification of parameters
t
in estimation, we impose additional restrictions; for example, we require that s(Y) be
t
diagonal.4
2Weassumethenondegeneracycondition,thattheinstantaneouscovariancematrixofthestatevariablesbe
full-rankforatleastsomevalueofthestatevector.
3Throughout,‘‘existence’’referstotheexistenceofaweaksolution,and‘‘uniqueness’’referstouniquenessin
distribution.
4ThisnormalizationisoneofseveralthatDaiandSingleton(2000)use.Thequestionofwhichaffineprocesses
can be represented with a diagonal diffusionmatrix by a change of variables is addressed by Cheridito, et al.
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128 P.Cheriditoetal./JournalofFinancialEconomics83(2007)123–170
AlthoughinEq.(7),wecharacterize bond pricesasconditionalexpectations (under the
Q measure), in practice, bond prices are usually calculated as solutions to a partial
differential equation, which, for the affine models we consider here, is equivalent to a
system of Riccati-type ordinary differential equations (see Duffie, et al. 2003). The
Feynman-Kac theorem, which establishes the equivalence of the probabilistic problem
and the partial differential equation problem, is well known and frequently applied to
affine term structure models. However, its applicability to bond prices under some
familiesofaffinemodelshasbeenformallyjustifiedonlyrecently;seeLevendorskii(2004a)
fortheaffinediffusioncase,andLevendorskii(2004b)forthecaseofaffineprocesseswith
jumps. For general payoff functions, the applicability of the Feynman-Kac theorem
remains an open issue; for bond prices, Levendorskii (2004a) establishes sufficient
conditions for the applicability of the Feynman-Kac theorem for all models we consider.
GrasselliandTebaldi(2004)establishnecessaryandsufficientconditionsfortheexistence
of closed-form solutions to the partial differential equation (which, as we state above,
is equivalent to a system of Riccati-type ordinary differential equations) for affine
yield models.
Given a specification of mP(Y) and s(Y) such that a solution to Eq. (2) exists, we may
t t
define an equivalent probability measure
(cid:8) Z T 1Z T (cid:9)
Q¼exp (cid:3) lTðY ÞdWP(cid:3) lTðY ÞlðY Þdu P (12)
u u 2 u u
0 0
by specifying a market price of risk process l(Y) that satisfies the condition
t
(cid:4) (cid:8) Z T 1Z T (cid:9)(cid:5)
EP exp (cid:3) lTðY ÞdWP(cid:3) lTðY ÞlðY Þdu ¼1. (13)
u u 2 u u
0 0
It follows from Girsanov’s theorem that the process WQ ¼WPþRtlðY Þds is an
t t 0 s
N-dimensional Brownian motion under Q, and
dY ¼mQðY ÞdtþsðY ÞdWQ, (14)
t t t t
where mQ(Y) is given by
t
mQðY Þ¼mPðY Þ(cid:3)sðY ÞlðY Þ. (15)
t t t t
Numeroussufficiencycriteria,such asthose ofNovikovand Kazamaki(see,forexample,
Revuz and Yor, 1994) have been developed to show that a given stochastic exponential
satisfies Eq. (13). Dai and Singleton (2000) consider a simple market price of risk
specification,
lðY Þ¼sTðY Þl, (16)
t t
wherelisavectorofconstants.Byconstruction,thisspecificationensuresthatmQ(Y)isan
t
affine function of Y. When sT(Y) does not depend on Y, this market price of risk
t t t
(footnotecontinued)
(2005),whofindthatanyaffinediffusiondefinedonastatespace(0,N)M(cid:2)RN(cid:3)M (afteraffinetransformationof
thestatevariables)withMp1orMXN(cid:3)1canbediagonalized,withthetransformedprocesstakingvaluesinthe
samestatespace.TheyalsogiveexamplesofdiffusionswithM¼2andN¼4whosediffusionmatricescannotbe
diagonalizedbyaffinetransformation.However,inthispaper,weconsideronlyNp3,inwhichcaseatleastone
oftheconditionsMp1orMXN(cid:3)1isalwayssatisfied.Thus,ourassumptionofadiagonaldiffusionmatrixdoes
notresultinlossofgenerality.
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specification satisfies the Novikov criterion for any time interval [s, t]. The Novikov
criterion may also be satisfied for any time interval even when sT(Y) does depend on Y,
t t
depending on the values of the model parameters. However, in general, the Dai and
Singleton (2000) market price of risk specification only satisfies the Novikov criterion
on[s,t]whentos+eforsomepositivee.Thevalueofedependsonthemodelparameters,
not on s or Y. However, this form of local satisfaction of the Novikov criterion is
s
sufficientforthesatisfactionofEq.(13)(see,forexample,Corollary5.14inKaratzasand
Shreve, 1991).
Duffee (2002) refers to models with the market price of risk specification of Dai and
Singleton (2000) as completely affine, and introduces the more general class of essentially
affine models. The only constraint on the market price of risk specification in essentially
affine models can be characterized as follows: if a linear combination of state variables is
restricted, then the market price of risk of that linear combination must coincide with the
completely affine specification. By contrast, a linear combination of state variables that is
unrestrictedcanhaveanymarketpriceofriskconsistentwithaffinedynamicsunderboth
measures. For example, in the univariate model
dY ¼(cid:6)aPþbPY (cid:7)dtþsdWP, (17)
t t t
the single state variable is unrestricted, so l(Y) can be any affine function of Y. By
t t
contrast, in the univariate model
dY ¼(cid:6)aPþbPY (cid:7)dtþspffiYffiffiffiffiffidWP, (18)
t t t t
the single state variable is restricted. Consequently, the essentially affine market price of
p
ffiffiffiffiffiffi
risk for this model is l(Y)¼l Y for some constant l (with l¼0 possible). In other
t t
words,l(Y)isrestrictedtoensurethat,ifthevolatilityofanylinearcombination ofstate
t
variables approaches zero, the risk premium of that linear combination also approaches
zero. As withthecompletely affinemarket price ofrisk specification, theessentially affine
specificationsatisfiestheNovikovcriterionforsomefinitepositivetimeinterval(thesizeof
which depends on the model parameters, but not on the initial state vector), thereby
ensuring satisfaction of Eq. (13).
Our market price of risk specification, by contrast, imposes only those restrictions
necessary to ensure that the boundary nonattainment conditions are satisfied under both
the P and Q measures. In Section 3, we show that this requirement is sufficient to ensure
thatthemarketpriceofriskspecificationsatisfiesEq.(13).Notethattheessentiallyaffine
specification nests the completely affine market price of risk, and our specification, which
we refer to as the extended affine market price of risk, always nests both the completely
affine and essentially affine specifications. The completely and essentially affine
specifications coincide for some models, as do the the essentially and extended affine
specifications. However, the extended affine specification is always more general than the
completely affine specification.
While affine yield models are treated in a systematic way by Duffe and Kan (1996),
many other models in the literature, such as Vasicek (1977), Cox, et al. (1985), Balduzzi,
et al. (1996), and Chen (1996), are special cases of the general affine model. Dai and
Singleton (2000) note that for each integer NX1, there exist N+1 nonnested families of
N-factoraffineyieldmodels,anddeveloptheclassificationschemethatweusebelow.Each
affineyieldmodelcanbeplacedintoafamily,designatedA (N),whereNisthenumberof
M
state variables and M is the number of linearly independent b ,1pi,jpN with M
ij
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130 P.Cheriditoetal./JournalofFinancialEconomics83(2007)123–170
necessarilytakingonvaluesfrom0toN.TheA (N)modelcontainsMstatevariablesthat
M
are restricted. Each of these state variables follows a process similar to the Feller
square-rootprocess,exceptthatthedriftofonerestrictedstatevariablemaydependonthe
value of another restricted state variable. The remaining M(cid:3)N state variables are
unrestricted. The unrestricted state variables jointly follow a process similar to a
multivariate Ornstein-Uhlenbeck process, but with two modifications: both the drift
and the variance of an unrestricted state variable may depend on the values of the
restricted state variables.
For now, we take as given that our market price of risk specification is free from
arbitrage, and examine in detail each of the single-factor, two-factor, and three-factor
affine yield models that we estimate. In addition to specifying both the dynamics of the
state variables under both the P and Q measures and the definition of the interest
rate process, we specify any parameter restrictions needed to ensure existence of the
specified process or to ensure restricted state variables do not achieve their boundary
values. We also identify any restrictions needed to make sure that a model has a unique
representation.
2.1. Single-factor models
In a single-factor affine yield model, the interest rate process is specified as
r ¼d þd Y ð1Þ (19)
t 0 1 t
for some constants d and d . However, the state variable Y(1) can follow one of two
0 1 t
distinct types of diffusions, namely, the A (1) or A (1) model (as per Dai and Singleton,
0 1
2000). In the A (1) model, Y(1) follows the process
0 t
dY ð1Þ¼(cid:2)bPY ð1Þ(cid:3)dtþdWPð1Þ, (20)
t 11 t t
whereisWPð1ÞastandardBrownianmotionundertheobjectivemeasureP,andbP isan
t 11
arbitrary constant. Note that this process has no constant term in the drift, and the
diffusion coefficient has been normalized to one. These restrictions are not a loss of
generality, but rather a normalization that ensures a unique representation of the model.
Any process with an affine drift and constant diffusion can be transformed into a process
ofthistypebyanaffinetransformationofthestatevariable.Anobservationallyequivalent
interest rate model results by making an appropriate change to the d and d coefficients.
0 1
Under the measure Q, the process Y(1) can be written as
t
h i
dY ð1Þ¼ aQþbQY ð1Þ dtþdWQð1Þ, (21)
t 1 11 t t
whereWQð1ÞisastandardBrownianmotionunderQ.Theprocessexistsforanyvalueof
t
bP; furthermore, there is no finite boundary value (i.e., the process Y (1) an take on any
11 t
real value), and the boundaries at infinity are unattainable in finite time, regardless of the
parameter values. The market price of risk process is defined by
L ¼½sðY Þ(cid:4)(cid:3)1(cid:2)mPðY Þ(cid:3)mQðY Þ(cid:3)¼(cid:3)aQ(cid:11)bP (cid:3)bQ(cid:12)Y ð1Þ(cid:5)l þl Y ð1Þ. (22)
t t t t 1 11 11 t 10 11 t
InthecompletelyaffinemodelsofDaiandSingleton(2000),thel parameterisrestricted
11
to be zero. By contrast, in the essentially affine models of Duffee (2002), the l and l
10 11
parameters can take any values. Existence of the measure Q with either the completely
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affineoressentiallyaffinemarketpriceofriskspecificationfollowsfromsatisfactionofthe
Novikov criterion for a finite positive time interval, as we discuss above. For the A (1)
0
model, our market price of risk specification coincides with the essentially affine
specification, offering no further generality.
The A (1) model is based on the square-root process of Feller (1951). Under this
1
specification, the process Y(1) can be expressed as
t
dY ð1Þ¼haQþbQY ð1ÞidtþpffiYffiffiffiffiffiðffi1ffiffiffiÞffidWPð1Þ, (23)
t 1 11 t t t
whereWPð1ÞisastandardBrownianmotionundertheobjectivemeasureP.Notethatthe
t
diffusion term may be taken to be Y itself, rather than some affine function of Y, by an
t t
appropriatechangeofvariables,aswedescribeabove.Existenceofsuchaprocessrequires
only that aPX0. Y(1) is bounded below by zero; this state variable cannot achieve its
1 t
boundary value if 2aPX1. Under the measure Q, the process Y(1) can be written as
1 t
dY ð1Þ¼haQþbQY ð1ÞidtþpffiYffiffiffiffiffiðffi1ffiffiffiÞffidWQð1Þ, (24)
t 1 11 t t t
where WQð1Þ is a standard Brownian motion under the measure Q. The market price of
t
risk process is defined as
L ¼½sðY Þ(cid:4)(cid:3)1(cid:2)mPðY Þ(cid:3)mQðY Þ(cid:3)¼ aP1 (cid:3)aQ1 þ(cid:11)bP (cid:3)bQ(cid:12)pffiYffiffiffiffiffiðffitffiffiÞffiffi
t t t t pffiYffiffiffiffiffiðffitffiffiÞffiffi 11 11 t
t
(cid:5) l10 þl pffiYffiffiffiffiffiðffitffiffiÞffiffi. ð25Þ
pffiffiffiffiffiffiffiffiffiffiffi 11 t
Y ðtÞ
t
The completely affine and essentially affine specifications coincide for the A (1) model; in
1
both,thel parametercantakeanyarbitraryvalue,butthel parameterisrestrictedto
11 10
be zero. For each value of l , the Novikov criterion is satisfied for some finite positive
11
time horizon. We permit l to take on any value such that boundary nonattainment
10
conditions are satisfied under Q as well as P. This requirement can be expressed as
1
l10paP1 (cid:3)2. (26)
It is unclear whether this specification satisfies the traditional Novikov and Kazamaki
criteria; in Section 3, we use another method to show that it satisfies Eq. (13).
2.2. Two-factor models
Two-factor affine yield models have an interest rate process given by:
r ¼d þd Y ð1Þþd Y ð2Þ, (27)
t 0 1 t 2 t
wheretheprocess followedby Y(1) and Y(2)falls into one ofthree categories,the A (2),
t t 0
A (2), or A (2) family. The P-measure dynamics for the A (2) model are
1 2 0
"Y ð1Þ# "bP bP #"Y ð1Þ# "WPð1Þ#
d t ¼ 11 12 t dtþd t . (28)
Y ð2Þ bP bP Y ð2Þ WPð2Þ
t 21 22 t t
These dynamics reflect any change of variables necessary to ensure that the matrix s(Y)
t
is an identity matrix and the constant terms in the drifts of the state variables are zero.
ARTICLE IN PRESS
132 P.Cheriditoetal./JournalofFinancialEconomics83(2007)123–170
Evenwiththesenormalizations,however,theA (2)representationisnotunique,asanew
0
set of state variables can be formed by taking any orthogonal rotation of the old state
variables. Dai and Singleton (2000) choose the identification restriction bP ¼0, which
12
guarantees a unique representation whenever the two components of Y are not
t
independent, i.e., when the normalization does not also cause the bP parameter to be
21
zero.IfthenormalizationcausesbothbP andbP tobezero,thenareorderingofthestate
12 21
variable indices is also possible. This method of normalization also precludes b matrices
with eigenvalues that are complex conjugate pairs.5 Under the measure Q, the process
followed by Y is given by
t
"Ytð1Þ# "aQ1 # "bQ11 bQ12#"Ytð1Þ#! "WQt ð1Þ#
d ¼ þ dtþd . (29)
Ytð2Þ aQ2 bQ21 bQ22 Ytð2Þ WQt ð2Þ
No parameter restrictions are needed to ensure the existence of the process, or of the Q
measure. Furthermore, there are no finite boundaries and no additional boundary
nonattainment conditions. The market price of risk specification is
L ¼½sðY Þ(cid:4)(cid:3)1(cid:2)mPðY Þ(cid:3)mQðY Þ(cid:3)
t t t t
0 2aQ3 2bP (cid:3)bQ bP (cid:3)bQ 3"Y ð1Þ#1
1 11 11 12 12 t
¼@(cid:3)4 5þ4 5 A
aQ2 bP21(cid:3)bQ21 bP22(cid:3)bQ22 Ytð2Þ
"l # "l l #"Y ð1Þ#
10 11 12 t
(cid:5) þ . ð30Þ
l l l Y ð2Þ
20 21 22 t
Thecompletelyaffinemarketpriceofriskspecificationsrestrictsl ,l ,l ,andl tobe
11 12 21 22
zero.Theessentiallyaffinespecificationrelaxestheserestrictions,andallowsallsixmarket
priceofriskparameterstotakeonarbitraryvalues.Bothofthesespecificationssatisfythe
Novikov criterion for a finite positive time interval, thereby ensuring that the specified Q
measureexistsandisequivalenttoP.FortheA (2)model,ourspecificationcoincideswith
0
the essentially affine market price of risk, offering no further flexibility.
The P measure dynamics of the A (2) model are given by
1
"Ytð1Þ# 0"aP1 # 2bP11 0 3"Ytð1Þ#1
d ¼@ þ4 5 Adt
Ytð2Þ 0 bP21 bP22 Ytð2Þ
2pffiffiffiffiffiffiffiffiffiffiffi 3 2 3
Ytð1Þ 0 WQt ð1Þ
þ4 0 pffiaffiffiffiffiffiþffiffiffiffiffibffiffiffiffiffiffiYffiffiffiffiffiðffiffi1ffiffiÞffiffi5d4WQð2Þ5, ð31Þ
2 21 t t
wherea A{0.1}.ExistenceofthisprocessrequiresthataPX0andbPX0.TheprocessY(1)
1 1 21 t
isboundedfrombelowbyzero;theadditionalrestriction 2aPX11sneededtoensurethat
1
Y(1) does not achieve the boundary value. The dynamics under the measure Q for the
t
5Dependingonthenumberandthematuritiesofthebondyieldsobserved,identificationissuesmayarisewhen
someoftheeigenvaluesoftheslopematrixinthedriftarecomplex.SeeBeagleholeandTenney(1991).