Random Field and Affine Models for Interest Rates

57 Pages·2004·1 MB·English

Random Field and Affine Models for Interest Rates:
An Empirical Comparison
C. Alan Bester ∗
First draft: October 2003
This version: September 2004
ABSTRACT
Traditional affine models of the term structure are eminently tractable, but suffer from
empirical difficulties. Random field models offer greatflexibility in fitting the data, but are
widely considered non-implementable unless they are approximated by a low-dimensional
system. I develop a state-space estimation framework where both random field and affine
models can be estimated by MCMC using the same panel of forward rate data. I find that
random field models are much better able to fit the patterns of volatility and correlationin
a long historical sample of U.S. Treasury forwardrates.
Keywords: Term Structure of Interest Rates, Affine Models, Random Fields, Markov
Chain Monte Carlo
JEL classification: E43, G10, C51, C52
∗Graduate Schoolof Business, Universityof Chicago, e-mail: [email protected]. I thankTim Boller-
slev, Michael Brandt, Bjørn Eraker, George Tauchen, and seminar participants at Carnegie Mellon University,
UniversityofChicago,DukeUniversity,IowaStateUniversity,NorthwesternUniversity,UniversityofRochester,
University of Texas-Dallas, and University of Toronto for many helpful commentsand suggestions. I also thank
friends at Pension Research Institute, Inc. in Chapel Hill, NC for providing me with bond price data used in
this paper.
1. Introduction
ModerntermstructuremodelingbeganwithVasicek(1977)andCox,Ingersoll,andRoss(1985).
Their work was later extended to the broader affine class of models (Duffie and Kan, 1996),
which were classified in a convenient hierarchy by Dai and Singleton (2000). Affine models are
distinguished by the assumption that the spot rate and the instantaneous covariances of yields
are linear in a finite set of diffusive state variables. These models have enjoyed a long and
productive life in the finance literature, due in large part to their eminent tractability—they
offer convenient forms for bondprices, yields, andforwardrates, and areeasily adapted to price
interest rate derivatives (see Duffie, Pan, and Singleton, 2000). Unfortunately, affine models
suffer from potentially serious empirical shortcomings. Chan et al. (1992) observe that the
Vasicek and CIR models fail to capture the stochastic volatility in short-term interest rates.
Duffee (2002) and Dai and Singleton (2003a) find that the more recent versions of affine models
struggle to explain the dynamics of volatilities and risk premia in observed bond yields.1
Recent empirical work suggests that other aspects of the data are missed by popular affine
models. In a study of bond yields, Litterman and Scheinkman (1991) find that over 95% of the
variation in the data can be explained by the first three principal components. Most empirical
applications thus consider models with three diffusive factors. However, Jagannathan, Kaplin,
and Sun (2001) find that the joint dynamics of yields, caps, and swaptions are not captured in
low-dimensional affine models. Dai andSingleton (2003b) suggest that this problem arises from
the model-implied correlation structure; they find that the correlation between innovations in
non-overlapping intervals of the yield curve is much lower in observed data than a two or three
factor model would suggest.
Random field models offer a parsimonious and flexiblealternative to traditional factor mod-
els. Pioneered by Kennedy (1994, 1997) and Goldstein (2000), and extended by Santa-Clara
and Sornette (2001) and Kimmel (2003), random field models are driven by a continuum of
stochastic processes with drifts, diffusions, and cross-sectional correlations specified as smooth
1AlternativestotheaffineclassincludetheessentiallyaffinemodelsproposedbyDuffee(2002),thequadratic
modelsdiscussedbyLongstaff (1989), Beaglehole andTenney(1992), Constantinides(1992), andAhn,Dittmar,
and Gallant (2002), the nonlinear model of Ahn and Gao (1999), and the hybrid models considered in Ahn,
Dittmar, Gao, and Gallant (2003). Though promising, these models have not yet attained the popularity of
affine models.
1
functions of maturity. As a result, each bondprice, forwardrate, or yield has its own stochastic
shock, at least a portion of which may be independent of shocks to all others in the sample.
In the finite-factor setting, if the term structure is driven by an N-factor model, any observed
sample of bond prices, yields, or forwards may have covariance matrix of rank at most N. This
places severe restrictions on the covariance structures that can be produced by factor mod-
els. In random field models, a similar set of observed securities may have covariance matrix
of full rank, and the model-implied covariance structure is restricted only by mild regularity
conditions.
The flexibility offered by random field models comes at a price. First, as Collin-Dufresne
and Goldstein (2003) point out, random fields are widely viewed as non-implementable unless
they are reduced to a low-dimensional model. Previous empirical studies by Longstaff, Santa-
Clara, and Schwartz (2001a,b) and Han (2002) have estimated random field models using a
principal-components approach where the parameters of the field model are chosen to match
the first M components in the data. However, Kerkhof and Pelsser (2002) observe that field
models estimated in this manner are observationally equivalent to finite-factor models. Second,
consider a typical application where the parameters of a term structure model are to estimated
from a panel of k observed bonds, yields, or forwards. In a traditional N-factor model, a set
of k N nonredundant securities completes the market, and the parameters defining the risk
≥
neutral distribution can be identified. In random field models, because each point on the yield
curveexhibitsvariation independentofallothersinthesample,riskpremiacannotbeidentified
from a finite set of observed bond data. Furthermore, random field models will in general be
consistent with any shape of the yield or forward rate curve observed on a particular date.
Many popular applications of affine models, including analysis of pricing errors, out-of-sample
forecasting, and derivatives pricing, cannot be performed in a random field setting without
additional information.
In this paper, I aim to partially bridge the gap between affine and random field models.
I develop a unified estimation framework where both classes of models can be estimated by
Markov Chain Monte Carlo (hereafter MCMC) from the same historical data on forward rates.
Severalpreviousstudieshaveestimatedaffinemodelsusingastate-spacerepresentation.2 Ishow
2See e.g. Geyer and Fru¨hwirth-Schnatter(1996), Mikkelsen (2001), Lamoureux and Witte (2002), Chen and
Scott (2003), and Sanford and Martin (2003).
2
that this state space framework can accommodate random fields by changing the interpretation
of the observation equation. In this setting, random field models can be implemented without
relying on a finite-factor approximation.
I estimate both sets of models using a long historical sample of forward rate data and
compare their empirical performance. The relevant question becomes, can random field models
explain aspects of the data which affine models cannot? The answer is yes: I find that affine
models fail to fit the volatility and correlation structure of forward rates of different maturities.
In comparison, using the flexible series form for correlations I propose below, random field
models are able to reproduce these aspects of the data nearly perfectly.
The rest of the paper is organized as follows. In section 2, I briefly review both affine and
random field models of the term structure. The development of affine term structure models
(ATSMs) uses the canonical form of Dai and Singleton (2000), hereafter DS. The discussion
of random field models (RFTSMs) for forward rates follows Santa-Clara and Sornette (2001).
The class of random field models I estimate is dueto Kimmel (2003). Section 3 provides details
of the MCMC algorithms for estimation of both classes of models using a panel of forward rate
data. The data and results are discussed in section 4. Section 5 offers concluding comments
and suggests future work on random field models.
2. Term Structure Models
2.1. Affine Models
ATSMs begin by assuming the short rate r is a linear combination of N diffusive factors,
t
r = δ +δ′ X , (1)
t 0 X t
where δ R and δ RN. Here X is an N-dimensional diffusion process satisfying the
0 X t
∈ ∈
stochastic differential equation (SDE)
dX = (Θ X )dt+Σ S dW , (2)
t t t t
K −
p
whereW is a standard N-dimensional Brownian motion, Θ is a vector of parameters determin-
t
ingthe mean of X , and Σ and aresquarematrices of parameters determiningvolatilities and
t
K
thespeedofmeanreversion,respectively. ThecomponentsofX canbeclassifiedintwogroups:
t
3
square root or Cox, Ingersoll, and Ross (1985) processes, and Gaussian or Ornstein-Uhlenbeck
processes. In DS notation, an A (N) model has N diffusive factors, m of which have square
m
root dynamics and N m of which are Gaussian. Without loss of generality, partition the state
−
vector X = (Xsr)′,(Xou)′ ′ so that the m square root factors appear first. An admissible
t t t
ATSM then h(cid:0)as the form (cid:1)
dXsr sr 0 Θsr Xsr
t = K m×(N−m) t +Σ S dW , (3)
t t
 dXou   sr,ou ou  0 − Xou 
t K K (N−m)×1 t p
      
where S is a diagonal matrix that determines conditional volatilities,
t
[Xsr] 1 i m
[S ] = t i ≤ ≤ (4)
t ii  α +β′Xsr m < i N
 i i t ≤
For later convenience, I define 
0 I 0
m×1 m×m m×(N−m)
 α   β′ 
m+1 m+1
α = = ,
. .
 ..  B  .. 0(N−m)×(N−m) 
   
   
 αN   βN′ 
   
so that S = α+ X . Hereafter I will refer to Θ, ,Σ,α, as the factor parameters, as they
t t
B { K B}
govern the physical dynamics of X . Note the square root processes Xsr have support on Rm
t t +
while the Gaussian processes have support on RN−m. Thus the restriction in (4) that S only
t
dependon Xsr is necessary to keep conditional volatilities positive almost surely. Thefollowing
t
additional restrictions on the factor parameters ensure the positivity of Xsr:
t
m Θ > 0 and Θ > 0 1 i m
j=1Kij j j ≤ ≤ . (5)
P i,j < 0 1 i,j m,j = i
K ≤ ≤ 6
Theserestrictions giverisetoatradeoffinaffinemodels. For afixedvalueofN,modelswith
more square root factors (larger m) offer increased flexibility in modeling stochastic volatility,
because conditional volatility can only depend on the square root factors. Models with a larger
number of Gaussian factors are more flexible in fitting conditional correlations, because the
final restriction in (5) restricts the correlation between square root factors.
To close the model, assume that the market price of risk satisfies
Λ = λ S , (6)
t t
p
4
where λ is an n-vector of risk premia. This is to say that the stochastic discount factor M
t
satisfies
dM
t
= r dt Λ dW (7)
t t t
M − −
t
whereW isaBrownianmotionunderthephysicalmeasure. Iwillreferto λ,δ ,δ astherisk
t 0 X
{ }
neutral parameters, as they determine the dynamics of the stochastic discount factor (and thus
the equivalent martingale or “risk-neutral” measure). Defining P(X ,x) as the time-t price of
t
a $1 pure discount bond maturing at time t+x, Duffie and Kan (1996) show that bond prices
inherit the exponential-affine form
P(X ,x) = eA(x)−B(x)′Xt,
t
where A: [0, ) R and B : [0, ) RN. In addition, the factor coefficients A(x) and B(x)
∞ → ∞ →
can be shown to satisfy the Ricatti ODEs
dB(x) 1
= ˜′B(x) Σ′B(x)diag Σ′B(x) + δ (8)
X
dx −K − 2 B
dA(x) 1 (cid:2) (cid:3)
= Θ˜′˜′B(x) + Σ′B(x)diag Σ′B(x) α δ , (9)
0
dx − K 2 −
(cid:2) (cid:3)
with boundary conditions B(0) = 0 and A(0) = 0. Here ˜ and Θ˜ determine the dynamics
N×1
K
of X under the risk neutral measure and are given by3
t
˜ = +ΣΦ
K K
Θ˜ = ˜−1( Θ Σψ),
K K −
where Φ is a matrix whose ith row is λ times the ith row of and ψ is an N-vector with
i
B
ith entry λ α . In some cases (e.g., the one-factor Vasicek and CIR models), the solutions to
i i
(8)–(9) are available in closed form. Even for large-dimensional models, these ODEs can be
solved accurately and efficiently using numerical methods.
For future reference, I note the form of zero coupon yields and instantaneous forward rates
implied by ATSMs. The zero coupon yield curve is defined as
logP(X ,x) 1
y(X ,x) = t = A(x)+B(x)′X . (10)
t t
− x x −
(cid:0) (cid:1)
3Under the assumption (6), the factor process Xt is affine under the risk neutral measure Q with dynamics
given by
dXt = ˜(cid:16)Θ˜ Xt(cid:17)+Σ√StdW˜t,
K −
where W˜t is a standard Brownian motion under Q.
5
Fixing t < x < s, the time-t forward price for delivery at time t+x of a bond that matures at
time t+s is the ratio of zero coupon bond prices, P(X ,s)/P(X ,x). The associated forward
t t
rate is defined as
logP(X ,s)+logP(X ,x)
t t
F(X ,x,s) = − .
t
s x
−
Assuming A and B are continuously differentiable, the instantaneous forward rate f(X ,x) is
t
defined as
dA(x) dB(x)′
f(X ,x) = limF(X ,x,s) = + X . (11)
t t t
s↓x dx dx
Here f(X ,x) can be interpreted as the continuously compounded interest rate on a loan which
t
is agreed to at time t, paid out at time t+x, and repaid an instant later. For the remainder of
this paper, I will refer to f(,x) simply as forward rates.
·
The chief advantage of affine models is the convenient form for zero coupon bond prices,
yields, and derivatives. For particular values of the factors X and the factor and risk neutral
t
parameters, model-implied bond yields or forward rates can be quickly and efficiently deter-
mined by solving the ODEs (8)–(9) and then evaluating the appropriate equation (10) or (11).
This tractability facilitates econometric estimation, forecasting, and derivatives pricing.
Unfortunately,ATSMsfaceanumberofchallengesinempiricalapplications. Thetradeoffin
modelingstochasticvolatility versuscorrelationstructurewasnotedabove. Inaddition,because
the market prices of risk can depend only on the square root factors through S , there is a three
t
way tension between flexibility in specifying conditional volatility, conditonal correlation, and
time variation in risk premia. This tension is noted by Duffee (2002), who links restrictions on
risk premia with the inability of affine models to explain excess returns to long-term bonds. It
also plays a central role in Jagannathan, Kaplin, and Sun (2001), who document that affine
models calibrated to one type of interest rate derivative (e.g., caps) systematically misprice
other derivatives (e.g., swaptions).
2.2. Random Fields
I beginthediscussion of RFTSMswith apurposefullyheuristicpresentation of randomfields as
continuous multiparameter martingales. The naturalstarting pointis the Brownian sheet W ,
t,x
which is the generalization of Brownian motion to a two-parameter martingale. In randomfield
modelsof theterm structure,thastheusualinterpretation ascalendar time, whilexis thetime
6
remaining until maturity. A more rigorous treatment of random fields, including generalization
of Itˆo’s stochastic integral to multiparameter martingales, is presented in Khoshnevisan (2002)
and Nualart (1995).4
Standard one-dimensional Brownian Motion is often constructed as
t
W = W + ǫ ds, (12)
t 0 s
Z0
where ǫ is a scalar white noise process, i.e.
t
E[ǫ ] = 0 (13)
t
cov[ǫ ,ǫ ] = δ(t s).
t s
−
The white noise process in (13) is defined over a one-dimensional domain, t [0, ). One can
∈ ∞
similarly define a two-dimensional white noise process ǫ over the domain (t,x) [0, ) R
t,x
∈ ∞ ×
by
E[ǫ ] = 0 (14)
t,x
cov[ǫ ,ǫ ] = δ(t s)δ(x y).
t,x s,y
− −
Thinking of x as a spatial variable and with t denoting time as usual, this process is “spatial-
temporalwhitenoise”. Onecanthinkofthisasacollection ofindependentwhitenoiseprocesses
indexed by x; for any fixed x and x , ǫ and ǫ are independent white noise processes.
0 1 t,x0 t,x1
The Brownian sheet W can be similarly defined by
t,x
1 x t
W = W + ǫ dsdz. (15)
t,x 0,0 s,z
√x
Z0Z0
This construction is really no more complicated than (12)—for any fixed values x and x , the
0 1
t t
inner integrals ǫ ds and ǫ ds generate two independent standard Brownian motions,
0 s,x0 0 s,x1
both continuouRs in t. ThinkinRg discretely, the outer integral adds the values of x independent
Brownian motions together and scales them by √x. Thus for any fixed x , W is a standard
0 t,x0
Brownian motion. It is tempting to say that W is simply a collection of Brownian motions
t,x
indexed by the spatial variable x. However, this construction delivers one more important
4The stochastic calculus of multidimensional processes was developed in the probability literature by Wong
andZakai(1974) andCairoli andWalsh(1975). SamplepathpropertiesofrandomfieldsaredevelopedinAdler
(1981).
7
property: because the Brownian sheet is constructed by integrating over x, W is continuous
t,x
in both x and t. A simulated Brownian sheet is shown in Figure 1.
In physical terms, Brownian motion W(t) represents the position at time t of a particle
subject to random changes in its velocity. One can similarly think of W(t,x) as the motion of
a stochastic string. Viewed in isolation, each point on the string follows a Brownian motion.
However, even though each pointonthe stringmoves randomly, thestringdoes notbreak apart
over time. Notice that for any x < y, the standard Brownian motions W(t,x) and W(t,y) are
not independent; however, each W(t,y) does contain some variation independent of W(t,x).
One can verify that
x y
corr[W(t,x),W(t,y)] = ∧ . (16)
x y
r ∨
Notice that corr[W(t,x),W(t,y)] = 1 only when y and x are equal.5
This construction of the Brownian sheet can be modified to generate a broad class of two-
parameter processes called random fields. A random field Z is distinguished by the spatial
t,x
correlation structure of its increments. Using the differential notation d to denote increments
t
inthet-direction (to distinguishfromincrements inthespatial direction), definethecorrelation
function c(x,y) as
c(x,y) = corr[d Z ,d Z ]. (17)
t t,x t t,y
Note that the correlation structure of the field increments does not depend on t. Equation (17)
may not make mathematical sense for some functions c. The following definitions summarize
the properties of admissible correlation functions and the corresponding random fields.
Definition 1 The square-integrable function c(x,y) :[0,L] [0,L] R is an admissible corre-
× →
lation function if and only if, for all (x,y) [0,L] [0,L],
∈ ×
(C1) c(x,x) = 1.
(C2) c(x,y) 1.
| | ≤
(C3) c(x,y) = c(y,x).
5To derive (16), write
√x 1 Z yZ t
W(t,y)= W(t,x)+ ǫ(s,u)dsdu.
√y √y
x 0
We use the standard probability notation x y = min(x,y) and x y = max(x,y). Notice that because we
∧ ∨
assumed x<y,we could havewritten corr[W(t,x),W(t,y)]=qx.
y
8
(C4) c(x,y) is positive semidefinite.6
The random fields in (17) can be constructed similarly to the Brownian sheet. Given that
c(x,y) is an admissible correlation function, Z has several important properties.
t,x
Definition 2 The process Z is an admissible random field if, for all (t,x) [0, ) [0,L],
t,x
∈ ∞ ×
(RF1) The mappings t Z and x Z are continuous.
t,x t,x
→ →
(RF2) E[d Z ] = 0.
t t,x
(RF3) var[d Z ] = dt.
t t,x
(RF4) Z(t,x) has correlation structure given by (17) where c(x,y) satisfies (C1)-(C4).
Each of these properties will prove important in building random field models of the term
structure. (RF1) ensures that forward rates and yields will be continuous in calendar time
t and time to maturity, x. (RF2) and (RF3) imply that for each x, Z is a unit-variance
t,x
martingale. Together, (RF2)–(RF4) imply that admissible random fields are Markovian.
An admissible random field Z can be constructed as follows.7 First, given an admissible
t,x
correlation function c(x,y), there exists a symmetric function g(x,y) such that
∞
g(x,y)g(x,z)dz = c(x,y) (18)
Z0
∞
[g(x,z)]2 dz = 1. (19)
Z0
The admissible random field Z then satisfies
t,x
L t
Z = Z + g(x,z)ǫ dsdz,
t,x 0,0 s,z
Z0 Z0
or equivalently,
L
d Z = g(x,z)ǫ dz dt. (20)
t t,x t,z
(cid:20)Z0 (cid:21)
Inequation(15),theBrowniansheetW isformedbyscalingandaveragingindependentBrow-
t,x
nian motions. The time increments d W are thus a simple average of white noise processes
t t,x
6For any ψ L2[0,L],
∈ Z LZ L
ψ(x)c(x,y)ψ(y)dxdy 0
≥
0 0
7This construction is due to Santa-Clara and Sornette (2001). The authors show existence of g(x,y) in
equations (18)–(19) in AppendixB of their paper.
9