Approximating Risk Premium on a Parametric Arbitrage-Free Term Structure Model

44 Pages·2015·0.62 MB·English

Approximating Risk Premium on a
Parametric Arbitrage-Free Term
Structure Model
*
Caio Almeida**
Kym Ardison***
Daniela Kubudi****
Abstract
In this paper we approximate the risk factors of a polynomial arbitrage-free dynamic
termstructuremodelbyrunningasequentialsetoflinearregressionsindependentacross
time. Thisapproximationavoidsthecostofafulloptimizationprocedureallowingfora
simplemethodtoextracttheriskpremiumembeddedininterestrateassets. Closed-form
bond pricing formulas provide interpretation of each source of aggregate risk as known
term structure movements. Assuming, for illustrative purposes, an economy with three
sourcesofaggregaterisk(level,slopeandcurvature),wetestthevalidityofourapproxi-
mationadoptingadatasetofBrazilianzerocouponinterestrates. Thenewmethodology
generates accurate parameters, standard deviations and risk premium dynamics when
compared to the exact dynamic model. Moreover, an empirical illustration comparing
the out-of-sample forecasting performance of the exact model, our approximation, and
a standard Gaussian affine model against the Random Walk, reveals that the proposed
approximationobtainsfavorableforecastingresultsprovidingareliablesubstitutetothe
exact estimation method with a much smaller computational time.
Keywords: Term Structure of Interest Rates, Parametric Models, Affine Models, Cross
Sectional Estimation, Time Series Analysis, Forecasting.
JEL Codes: C1, C5, G1.
*Submitted in April 2015. Revised in April 2015. We thank Marcio Garcia (discussant)
andparticipantsattheXIIBrazilianMeetingofFinanceforusefulcomments. Thefirstauthor
acknowledgesfinancialsupportfromCNPq-Brazil.
**GraduateSchoolofEconomics,FGV/EPGE,RiodeJaneiro,Brazil.
E-mail: [email protected]
***GraduateSchoolofEconomics,FGV/EPGE,RiodeJaneiro,Brazil.
E-mail: [email protected]
****GraduateSchoolofEconomics,FGV/EPGE,RiodeJaneiro,Brazil. Addressforcorrespon-
dence: PraiadeBotafogo190,11o andar,22250-900,Botafogo,RiodeJaneiro.
E-mail: [email protected]
BrazilianReviewofEconometrics
v. 34,no 2,pp. 203–246 November2014
CaioAlmeida,KymArdisonandDanielaKubudi
1. Introduction
Parametric arbitrage-free term structure models are dynamic models with
closed-form bond pricing formulas that explicitly depend on the risk factors of
theunderlyingeconomy. Althoughtheoreticalresultsonsuchmodelswerederived
aboutfifteenyearsago,correspondingempiricalimplementationsandapplications
have only recently gained more attention.1
Among some interesting features, parametric term structure models are useful
and straightforward to understand given that factor loadings are known before
model estimation. This a priori knowledge also brings some advantages for para-
metric term structure models when compared to traditional dynamic affine and
quadraticones.2 Forinstance,closed-formformulasforbondpricesguaranteethat
eachsourceofriskintheeconomyhasinterpretationasaknownmovementofthe
term structure.3 Nonetheless, perhaps the most important advantage is that this
featureallowsresearcherstobetterdisentangletheroleofcross-sectionalandtime
series information on the identification of model parameters.
Inthispaper,werestrictourattentiontoaparticularparametricarbitrage-free
model, the Legendre Dynamic Term Structure Model (Almeida, 2005). It param-
eterizes the term structure of interest rates as a linear combination of Legendre
polynomials, with coefficients of the linear combination representing the model
state vector. We propose a simple and fast method to approximate the model so-
lutionwithouthavingtofullyoptimizethedynamicmodel. Byconsideringalinear
regressionformulationinourapproximation, themethodisfasttoimplementand
readily delivers risk premium implicit in fixed income securities.
By exploring closed-form formulas of bond prices we are able to separate the
estimationprocedureintwosequentialsteps: Cross-sectionandtime-series. Inthe
cross-section step, we propose to approximate the model state vector by running,
for each date, a linear multiple regression of observed zero-coupon yields on a
setofpredeterminedtermstructuremovementswhoseloadingsarerepresentedby
Legendrepolynomials. Inthesecondstep,usingthefactthatdeterministicfactors
are in general small for the Legendre model,4 we run a Vector Autoregressive
(VAR)regressionoflagoneonthetime-seriesoftheapproximatedstatevectorto
estimate models’ driving parameters under the physical probability measure.
Separating the estimation in two stages allow us to come up with our approxi-
mation for the risk premium. After running the two pairs of regressions to obtain
both the approximated state vector and estimates for model parameters under
1For theoretical papers see Bjork and Christensen (1999), Filipovic (1999, 2000a,b, 2002),
BjorkandLanden(2002),SharefandFilipovic(2004)andAlmeida(2005). Forempiricalappli-
cations,seeAlmeidaandVicente(2008)andChristensenetal.(2011).
2ForaffinemodelsseeDuffieandKan(1996),forquadraticseeAhnetal.(2002).
3Incontrast,theloadingsofeachsourceofriskinusualdynamicmodels,onceestimatedhave
to be associated with approximations for the level, slope, curvature, and other term structure
movements.
4AlmeidaandVicente(2008)documentedthisfeaturewhenconsideringU.S.data.
204 BrazilianReviewofEconometrics 34(2)November2014
ApproximatingRiskPremiumonaParametricArbitrage-freeTermStructureModel
thephysicalmeasure,riskpremiumisbackedoutfromasimpleanalyticalformula
thatconnectsthosetwoprobabilitymeasuresasdoneintraditionaldynamicaffine
models.5 As a byproduct, once all risk-neutral parameters are known, the model
can be readily used in financial applications such as risk management procedures
or pricing interest rate options.6
In the empirical section we apply the described methodology to a database of
Brazilian zero coupon yields adopting an arbitrage-free Gaussian Legendre poly-
nomial model with three risky stochastic factors. Three versions of the model
are implemented: the exact dynamic model, our approximation based on regres-
sions,andacombinationofourapproximationmethodwithamaximumlikelihood
approach (henceforth mixed likelihood approach).
The main motivation for seeking an approximation method, as already men-
tioned, relies on the high computational cost involved when performing a full
optimization process. With only three stochastic factors and imposing some iden-
tification restrictions7 our exact method runs an optimization algorithm with 21
unknown parameters. On the other hand, using the mixed likelihood approxima-
tion leads to an optimization procedure8 with only 9 parameters to be estimated,
considerably reducing the computational cost.9
Numerical results show that parameter estimates as well as their standard de-
viations for all three methods are relatively close to each other, especially when
comparing the exact method with mixed likelihood approach. In particular, for
these last two implementations, the same subset of parameters is significant at a
95% confidence interval. In addition, as can be noted in Figure 6, risk premium
time series implicit in bonds of different maturities for all three versions of the
model estimation move on the top of each other for a large part of the sample,
and are very similar on the remaining sub-sample. This result reassures that the
approximation works as a good substitute for the original more complex imple-
mentation of the dynamic model. Moreover, a Monte Carlo experiment presented
in Section 5 confirms the validity of the two versions of the approximation.
Computationalcostsofimplementingaffinemodelscanbeaconsiderableprob-
lem when dealing with model estimation a repeated number of times. Thus, fore-
casting schemes with rolling windows provide a perfect illustration of the applica-
5Due to the parametric nature of the model, all risk neutral parameters, except for the
diffusion of the state vector, are known a priori. Since the diffusion is a common parameter to
bothphysicalandrisk-neutralmeasures,itisestimatedwiththeVARregression.
6Onthecurrentpaper,wefocusinshowingthattheproposedapproximationisaccurateand
especially useful in empirical applications that demand repeated estimation of the model, like
forecastingschemes.
7Forinstance,therestrictionsappearinginDaiandSingleton(2000).
8Ifweimposethesamerestrictionsasintheexactmethod.
9Running the exact method with yields with seven different maturities and time series of
1762 daily data can take as long as 150 seconds, while the average time in the Monte Carlo
experiment for the mixed likelihood approach is about 50 seconds. Our fastest approximation
basedontwo-stepregressionstakes(onaverage)only0.5secondstobecomputed.
BrazilianReviewofEconometrics 34(2)November2014 205
CaioAlmeida,KymArdisonandDanielaKubudi
bility and possible gains of adopting our methodology. Building on Duffee (2002)
weproposeaforecastingexercisebasedonaseriesofout-of-sampleforecasts,which
demand model re-estimation. Adopting essentially affine risk premium10 for the
three versions of the Legendre model (exact, mixed likelihood, and two-step re-
gression) and a traditional essentially Affine three factor model, for comparison
purposes, we predict future yields 1-week and 1-month ahead.
Results coming from this procedure show that with essentially affine premia
both the traditional affine model and Legendre model with exact estimation out-
perform the Random Walk for most maturities and forecasting horizons, corrobo-
ratingthepositiveresultsobtainedbyDuffee(2002)withessentiallyaffinemodels.
More interestingly however, is the fact that the approximation methods also out-
performtheRandomWalk, oftengeneratingresultsbetterthantheexactmethod
itself,withtheadvantageofhavingamuchsmallerestimationtimewhencompared
to either the exact Legendre model or the traditional affine one.11
Ourworkisrelatedtorecentpapersthatofferfastandefficientmethodsforthe
estimation of Gaussian arbitrage-free term structure models. Joslin et al. (2011)
suggestatwo-stepestimationprocedureforGaussianaffinetermstructuremodels
thatconsistsonacombinationofrunningordinaryleastsquaresregressionsunder
thephysicalprobabilitymeasurecombinedwithlikelihoodmaximizationmethods
to obtain risk-neutral parameters on a second step. Similarly Hamilton and Wu
(2012) provide a new estimation methodology based on the minimization of chi-
square distances. Although applicable to a more general class of Gaussian models
than ours, both methods still rely on optimization processes to estimate part of
the parameters while our approximation relies on running only two sets of linear
regressions.
Finally, Adrian et al. (2013) have the closest work related to ours. They sug-
gest using a three-step linear regression estimator to extract the risk premium of
Gaussiantermstructuremodelswithobservablepricingfactors. Incontrast,inour
method,sincewebuildthepricingfactorsbasedonLegendrepolynomialswedon’t
need observable pricing factors to exist. This has important consequences, in par-
ticular when analyzing interest rate markets that contain coupon-bearing bonds.
In such cases, by construction, Adrian, Crump and Moench approach needs ex-
ternal observable factors to be used as sources of risk since their method is not
designed to extract risk factors from the analyzed market, except if it is a zero
coupon market and principal component analysis can be applied. On the other
10AssuggestedbyDuffee(2002).
11In the forecasting exercise that we propose there is a need to re-estimate all versions of
the model 1005 times, each with a fixed window-size equal to 730 days. By making educated
guesses,thatis,usinginitialconditionsinonedayequaltothesolutionfromthepreviousday,it
tookus18hourstoestimatetheLegendremodelwiththeexactmethod,8hourswhenadopting
themixed-likelihoodapproach,andlessthanfiveminuteswhenadoptingthetwo-stepregression
scheme. Asaparameterofcomparison,alsousingeducatedguessesittook26hourstoestimate
thetraditionalthree-factoraffinemodelconsideringthewholeforecastingexercise.
206 BrazilianReviewofEconometrics 34(2)November2014
ApproximatingRiskPremiumonaParametricArbitrage-freeTermStructureModel
hand, since we have a parametric arbitrage-free model, we can extract risk factors
by using the model parametric nature and verify the prices of risk of such factors
by using the model arbitrage-free nature, even in markets with coupon-bearing
bonds.12
2. The Legendre Polynomial Model
ThissectionisdedicatedtoderivetheLegendrepolynomialmodelinadynamic
arbitrage-freeenvironmentandpresentthealgorithmtocomputetheexactversion
of the model. At the end of the section, we motivate our approximation methods
and explicitly show how to implement it.
2.1 The dynamic model
Theunderlyinghypothesisthatdrivesthismodelisthattheinterestrateterm
structure can be modeled as a linear combination of Legendre polynomials given
by:
(cid:88) 2τ
R(t,τ)= X P ( −1), ∀t (1)
t,n n l
n≥0
whereτ denotestimetomaturity, P (.)istheLegendrepolynomialofdegreen,13
n
X indicate each factor driving the term structure movements, l denotes the
t,n
longest maturity of a bond in the market considered, and t represents the time
period.
To incorporate this static set up within a dynamic one, one must assume the
existence of a probability space (Ω,F,Q), with Q being an equivalent martingale
measure under which discounted bond prices are martingales. The existence of
such measure guarantees absence of arbitrages in the market, in a dynamic sense
(see Duffie, 2001). Within this probability space, we assume that there is an N-
dimensional Brownian Motion W∗ and we refer to X as the state vector whose
t t
uncertainty is driven by W∗.
t
Given the Legendre polynomial structure, the martingale condition under Q
is guaranteed by imposing restrictions on the Stochastic Differential Equations of
the factors, X .14
t
12In this paper, however, we concentrate on the zero-coupon case, keeping in mind that the
methodology can be directly extended to a market with coupon bearing bonds. The exten-
sion should be based in Almeida et al. (2003) who prove that we can build synthetic principal
componentswithLegendrepolynomialsinmarketswithunobservabletermstructures.
1 dn
13Pn(x)= 2nn!dxn(x2−1)n.
14As pointed out by one of the referees, the martingale condition doesn’t guarantee static
no-arbitragesinceinstantaneousforwardratescanbenegativeintheGaussianLegendremodel.
Notehowever,thatthisistrueforthewholearbitrage-freeaffineliteratureinitiatedbyDuffieand
Kan (1996) since the yield curve is linear in the stochastic factors of the model. The only case
in this literature where instantaneous forward rates will be necessarily positive is when factors
followFellerprocesses. Henceforth,weusethetermarbitrage-freeformodelsinadynamicsense.
BrazilianReviewofEconometrics 34(2)November2014 207
CaioAlmeida,KymArdisonandDanielaKubudi
Although the original static Legendre model considers infinite polynomials,
for practical purposes we restrict our attention here to the finite case with N
polynomial factors.15 Under this finiteness restriction, the price of a bond with
maturity τ at time t becomes:
P(t,τ)=e−τ(cid:104)(cid:80)n(N=−01)Xt,nPn(2lτ−1))(cid:105), (2)
The only ingredient missing in the model is the dynamics of the factors X ,
t
whom drive the term structure movements. The underlying hypothesis is that
their dynamics are governed by the following Stochastic Differential Equation:
dX =µQ(X )dt+σ(X )dW∗ (3)
t t t t
Thus, given the model assumptions, the question that must be answered is:
Which restrictions µQ(X ) and σ(X ) should satisfy in order to guarantee that
t t
discounted bond prices are martingales? Fortunately, this problem was solved in
Almeida (2005).16 Writing bond prices as a function of the forward rate curve,
applying the Heath et al. (1992) methodology and Ito’s formula, Almeida derived
the martingale drift restriction that the state vector should satisfy under Q.
OneimportantaspecthighlightedbyAlmeida(2005)isthatinaN-dimensional
Brownian setting, the Dynamic Legendre Model with N factors will only have
[N]17 trulystochasticfactors, i.e., factorsdirectlydrivenbytheBrownianMotion
2
vector, with diffusion coefficients differing from zero. The other N −[N] factors
2
areconditionallydeterministicfactors,i.e.,theyhavenulldiffusioncoefficientsand
their dynamics are driven exclusively by the true stochastic factors parameters
(and the time variable). More clearly, this means that the diffusion matrix σ(X )
t
can have at maximum a (cid:2)N(cid:3) × (cid:2)N(cid:3) sub-matrix with nonzero terms and thus
2 2
implyingthattheconditionallydeterministicfactorsarenotaffectbytheBrownian
motions.
Forpracticalpurposes,AlmeidaandVicente(2008)suggesttobypasstheissue
of conditionally deterministic factors by considering Legendre dynamic models
with 2k polynomials driving the term structure whenever the researcher believes
that k stochastic factors are sufficient to model the uncertainty of the yield curve.
Thus, in line with previous literature that advocates that three stochastic factors
aresufficienttoexplaintermstructuremovements,weadoptasixfactorLegendre
models in the empirical section. With this structure, our stochastic factors have
the usual interpretation of level, slope and curvature of the term structure.18
Thatis,amodelisarbitrage-freeifdiscountedbondpricesundertheriskneutralmeasureQare
martingales.
15In particular, in our empirical applications we will consider a model with six Legendre
polynomials,whichturnsouttobeinlinewithtraditionalthree-factorAffinemodels.
16Seeproposition(2)ofthepaper.
17wherefunction[.]representstheintegerpartoftheargument.
18LittermanandScheinkman(1991)andHeidariandWu(2003)applyingprincipalcomponent
208 BrazilianReviewofEconometrics 34(2)November2014
ApproximatingRiskPremiumonaParametricArbitrage-freeTermStructureModel
In this paper, we further restrict our attention to Gaussian Legendre models
whose diffusion function are constant and Brownian motions are independent of
each other, implying a diagonal variance-covariance matrix.19 With this restric-
tion, Equation (3) becomes:
dX =κQ(θQ−X )dt+ΣdW∗ (4)
t t t
where Σ is a diagonal matrix. Note that this assumption does not imply that
factors X will be uncorrelated. In fact, their correlation will come from κQ and
t
κ (see (20)) respectively under the risk-neutral and physical probabilities.20
In the remainder of this section, we specify the previous model for the six
factors case and derive the necessary martingale restrictions that guarantee the
absenceofarbitrage. GiventhetermstructureR(t,τ)in(1),itispossibletoshow
that the instantaneous forward rate is given by:21
∂R(t,τ)
r(t,τ)=R(t,τ)+τ (5)
∂dτ
Using equation (1) we can specialize (5) to the Legendre parameterization as
follows:
 
r(t,τ)= (cid:88)Xt,nPn(2lτ −1)+τ(cid:88)Xt,n∂Pn(∂2lττ −1) (6)
n≥0 n≥1
Observing Equation (6), note that the instantaneous forward rate can be
rewritten as a polynomial in the maturity variable τ:
3
(cid:88)
r(t,τ)= L (X ,X ,...,X )τn (7)
n t,0 t,1 t,N−1
n=0
whereeachL isalinearfunctionofvector[X ,X ,...,X ](cid:48). Usingtheforward
n 0 1 N−1
rate Equation (5) with R(t,τ) given in (1), and collecting terms in powers of τ as
in (7) we obtain an auxiliary state vector X˜, related to the forward rates by the
following equation:
N
r(t,τ)= (cid:88)X˜ τn−1 (8)
t,n
n=1
analysisshowedthatthreefactorsarestatisticallyadequatetocapturethevariabilityoftheterm
structuresofU.S.Treasurybondsandswapyields,respectively. Morerecentpapersadvocatein
favorofalargernumberoffactors(sayfourorfive)ifthegoalistoforecastfutureyields(Duffee,
2011)orfuturebondreturns(CochraneandPiazzesi,2005).
19NotethatdiffusionsthataredeterministicfunctionsoftimealsogenerateaGaussianmodel.
20Correlation coming through a drift term instead of diffusion is a feature that will become
clearerwhenwederivethediscreteversionof(20)in(26).
21Seeeq.(4)-(6)inAlmeida(2005).
BrazilianReviewofEconometrics 34(2)November2014 209
CaioAlmeida,KymArdisonandDanielaKubudi
This auxiliary state vector X˜ is used only to simplify the calculation of the
Ito’s formula when imposing the HJM martingale condition (details appear in
Almeida (2005), or Almeida and Vicente (2008)). Note, however, that recovering
the original state space vector X and its SDE from X˜ should be a simple task. In
fact, to obtain X from X˜, we only have to solve the linear system:
X˜ =L (X ),j =1,2,...,N. (9)
t,j j t
where each L ,j =1,2,...,N comes from (7).
j
GiventhatX˜ isonlyalinearcombinationofX andviceversa,onceoneknows
the stochastic differential equation driving one of them it is easy to derive the
others. That is, by imposing the martingale restriction to one of them the corre-
sponding restrictions to the other are automatically determined. For tractability
westartwithrestrictionsimposedtotheSDEofX˜. Afterweobtaintherespective
restrictions on the dynamics of X.
Using the fact that X has Gaussian affine dynamics and that linear combina-
tions preserve this property, X˜ will also present Gaussian affine dynamics:
dX˜ =µ˜Q(X˜ )dt+Σ˜dW∗ (10)
t t t
where Σ˜ is a N ×N matrix22 and µ˜Q(X˜ ) is an affine function of X˜.
t
To find the restriction we must impose to the drift and diffusion of X˜ we rely
on Almeida (2005), who proved that discounted bond prices are Q-martingales
under Legendre model whenever the following equality holds:
(cid:88)N (j−1)X˜ τj−2 =(cid:88)N (µ˜Q(X˜ )) τj−1−(cid:88)[N2](cid:88)[N2]Σ˜Σ˜T τj+k−1 (11)
t,j t j jk k
j=2 j=1 j=1k=1
DenotingΣ˜Σ˜T =H˜ ,andmatchingτ coefficientsin(11),weobtainanexplicit
0
expression for the drift of X˜.23
µ˜Q(X˜ ) =X˜
t 1 t,2
µ˜Q(X˜ ) =2X˜ +H˜
t 2 t,3 0,11
µ˜Q(X˜ ) =3X˜ + H˜0,12 +H˜
µ˜Q(X˜t)3 =4X˜t,4+ H˜02,22 +H˜0,21 (12)
t 4 t,5 2 0,31
µ˜Q(X˜ ) =5X˜ + H˜0,23 + H˜0,32
t 5 t,6 3 2
µ˜Q(X˜ ) = H˜0,33
t 6 3
22ForidentificationpurposeswefollowDaiandSingleton(2002)andchoosetohaveadiagonal
matrix Σ. Note, however that the diffusion coefficient Σ˜ of X˜ will be upper triangular since it
is a linear transformation of Σ by an upper triangular matrix L−1 (see Equation (16) and the
appendix).
23AstepbystepderivationcanbefoundinAlmeidaandVicente(2008).
210 BrazilianReviewofEconometrics 34(2)November2014
ApproximatingRiskPremiumonaParametricArbitrage-freeTermStructureModel
Given the restriction on the dynamics of X˜ and equation (9) we can derive
the same set of restrictions for X. After some algebraic manipulations expressing
r(t,τ)asapolynomialofdegreefiveinτ andcalculatingeachcoefficientforpowers
of τ we explicitly obtain the linear transformation of (9):
X˜ =LX (13)
t t
where L is the upper triangular matrix, thus invertible, presented in equation
(31) in the Appendix I. To further simplify our calculations let U and M be a
6×6 and 6×1 matrix the collect the terms multiplying X˜ and the constants
respectively (matrices are defined in (32) and (33) in Appendix I). We can than
rewrite equations (12) as a affine function of X˜:
µ˜Q(X˜ )=M +UX˜ (14)
t t
Since X =L−1X˜, the drift and diffusion of X can be expressed as follows:
µQ(X )=L−1µ˜Q(X˜ )=L−1µ˜Q(LX )=L−1M +L−1ULX (15)
t t t t
σ(X )=Σ=L−1Σ˜ (16)
t
Equation (15) provides the functional relationship between κQ,θQ and the
matrices L,M,U:
κQ =L−1UL
(17)
κQθQ =L−1M
Equation (17) represents the two restrictions that the risk neutral drift in the
Legendre model should satisfy to avoid arbitrages. The first makes κQ predeter-
mined before model estimation (see Eq. (36)). This is a restriction coming from
the polynomial nature of the model. The second is a HJM-type restriction relat-
ing drift and volatility: The product κQθQ should be a specific function of the
volatility process ΣΣT, since M is a function of Σ (see Eq. (37)).
AnimportantpointtobenotedisthatX ,X ,andX aredeterministicfactors
4 5 6
under the Gaussian case. This is a consequence of the fact that their dynamics
do not depend on the Brownian motion vector, and their drifts do not depend on
the first three components of the state vector. In this case, we can write down
explicitly their dynamics as functions of time and volatility parameters Σ (see
Appendix II).
2.1.1 Market prices of risk and risk premia
Theelementthatconnectsfactorsdynamicsundertheriskneutral(orpricing)
measure Q and the physical measure, which we denominate P, is the market
BrazilianReviewofEconometrics 34(2)November2014 211
CaioAlmeida,KymArdisonandDanielaKubudi
price of risk, or in technical terms, the instantaneous volatility of the state price
deflator.24
Duffee (2002) and Dai and Singleton (2002) tested different specifications of
affine processes using U.S. treasury bonds data and concluded that a flexible time
varying risk premia, with prices of risk potentially depending on all risk factors
drivinginterestratesuncertainty,isthemostappropriatetocaptureinterestrates
dynamics.25
Based on their results and since we are restricting the Legendre model to the
affine class, we adopt the essentially affine risk premia of Duffee (2002) implying
the following form for the market prices of risk:
Λ =λ −λ X , (18)
t 0 X t
where λ is a N×1 vector, λ is a N×N matrix. Note however that since three
0 X
ofourfactorsaredeterministicvectorλ andmatrixλ willcontainonlyasubset
0 X
of nonzero entries (N/2 in fact). This choice of risk premium is also useful since
it allow us to have closed formulas for the discrete version of the state variables
under the physical probability P.
ThemarketpricesofriskallowustorelatetheBrownianMotionvectorsunder
the probability measures P and Q:
(cid:90) t
W∗ =W + Λ ds (19)
t t s
0
where, by Girsanov’s theorem, W is an N-dimensional Brownian Motion under
t
P.
Moreover, while the risk neutral dynamics of the state vector appears in (4),
whenwerestrictmarketpricesofrisktobeessentiallyaffine,itsphysicaldynamics
should be expressed by:
dX =κ(θ−X )dt+ΣdW (20)
t t t
After some algebra using the market prices of risk we can find the relationship
between the drift term under the probabilities measures P and Q as follows:
κ=κQ+Σλ (21)
X
 λ1 
0
.
κθ =κQθQ+Σ .  (22)
 . 
λN
0
24Thestatepricedeflatorisastrictlypositiveprocesswiththepropertythatitturnsdeflated
price processes into martingales under the physical measure. Absence of arbitrage implies the
existenceofatleastonestatepricedeflator(Duffie,2001).
25For a more detailed discussion on the importance of time varying risk premium we refer
toseveralpapersinthisliterature: DaiandSingleton(2002),Duffee(2002),andCochraneand
Piazzesi(2005).
212 BrazilianReviewofEconometrics 34(2)November2014