Mathemciricd
Finatiw,
Vol.
6.
No.
4
(October
l996),
379406
A YIELD-FACTOR MODEL
OF
INTEREST RATES
DARRELL
DUFFIE
Graduate
School
of
Busine.ss, Stanford University
RLJI
KAN‘
First
Boston
Corprution,
New
Ynrk
This paper presents
a
consistent and arbitrage-free multifactor model of the term structure of interest
rates in which yields at selected fixed maturities follow
a
parametric multivariate Markov diffusion
process with “stochastic volatility.” The yield
of
any zero-coupon bond is taken
to
be
a
maturity-
dependent affine combination of the selected “basis” set
of
yields. We provide necessary and sufficient
conditions on the stochastic model
for
this affine representation. We include numerical techniques
for solving the model,
as
wcll
as
numerical techniques for calculating the prices of term-structure
derivative prices. The case of jump diffusions
i\
also
considered.
I.
INTRODUCTION
This paper defines and analyzes a simple multifactor model of the term structure of interest
rates.
The factors of the model are the yields
X
=
(XI,
Xz,
.
.
.
,
X,l)
of zero-coupon
bonds of
n
various fixed maturities,
Itl,
r2.
.
,
.
,
t,,}.
For
example, one could think of
the current five-year (zero-coupon) yield as a factor. The yield factors form a Markov
process, to be described below, that can be thought of as a multivariate version
of
the
single-factor model of Cox, Ingersoll, and
Ross
(1
98%).
As
opposed
to
most multifactor
term structure models, the factors (Markov state variables) are observable from the current
yield curve and their increments can have an arbitrarily specified correlation matrix. The
model includes stochastic volatility factors that are specified linear combinations of yield
factors. Discount bond prices at any maturity are given as solutions
to
Ricatti (ordinary
differential) equations, and path-independent derivative prices can be solved by, among
other methods, an alternating-direction implicit finite-difference solution of the “usual”
partial differential equation (PDE). Fully workcd examples of solutions to these Ricatti
equations and
PDEs
are included.
Our yield model is “affine” in the sense that there is, for each maturity
t,
an affine function
Y,:
R”
+
R
such that, at any time
t,
the yield of any zero-coupon bond of maturity
t
is
Y,(X,).
Indeed, ruling
out
singularities, cssentially any
n
yields would serve as the factors,
and given the imperfections of any model, it is an empirical issue as to which
IZ
yields will
serve best as such. Likewise, because of linearity, the Markov state variables can be taken
to be forward rates
at
given maturities,
so
that the model can be viewed as
a
multifactor
Markov parameterization
of
the Heath, Jarrow, and Morton (HJM)
(1992)
model. In fact,
Frachot and Lesne
(1993)
have extended
our
model to the
HJM
setting. One could
also
take
‘We are gruteful fordiscussions with Ken Singleton,
Boh
Litterman, Antoine Conre, Nicole
El
Karoui. Vincent
Lacoste. Jeremy Evnine, Antoine Frachot, Henri
Pagk
Jean-Philippe Lesne, Fischer Black, Ayman Hindy. George
Pennnchi, Rob
Bliss,
Prasad Nannisetty, Stan Pliska,
Chri\
Rogers. Oldrich Vasicek,and especially to
a
referee for
pointing out an error in
an
earlier version.
Mamrsi~ript
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1996
Blackwell Publisher\,
238
Main Street. Camhridgc. MA
02142,
USA, and
108
Cowley Road, Oxford,
OX4
IJF.
UK.
380
DARRELL
DUFFIE AND
RUI
KAN
specified linear combinations of zero-coupon yields, such as the slope of the yield curve,
or even derivatives of the yield curve with respect to maturity at
a
given point.’
Special cases
of
our model are those of Chen
(1995),
Chen and Scott
(I
992),
Cox,
Ingersoll, and
Ross
(CIR)
(198%)
(in
its multivariate form), Heston
(1991),
El
Karoui and
Rochet (1989), Jamshidian
(1
989,
199
I,
I992), Langetieg
(1
980), Longstaff and Schwartz
(1
992),
and Pennachi
(1
99
1
).
In all of these other earlier models, the state-variable processes
are treated as “shocks” of various kinds that are not necessarily designed to be observable
from the current yield curve. After solving the models for the term structure, however, the
yield at any given maturity
r
can be seen to be a t-dependent affine function of the underlying
state variables. Given a set of maturities equal
in
number with the underlying factors, one
can therefore typically (that
is,
when the coefficient vectors defining the corresponding
affine forms are linearly independent) perform
a
change of basis under which the state
variables are yields at these various fixed maturities,
as
in our model. This idea has been
pursued
by
Pearson and Sun (1994) and by Chen and Scott (l993a), who have recently and
independently estimated
a
special case, based on a multifactor version of Cox. Ingersoll,
and Ross
(198Sa),
by performing just such a change of variables. Our model unifies and
strictly extends these affine models to the maximum possible degree, and fully exploits the
idea of using yields
as
state variables.
Empirical studies of multifactor models in our “aftine yield” setting include those
of
Brown and Schaefer
(1993),
Chen and Scott
(1992,
1993b), Duffie and Singleton
(1995),
Frachot, Janci, and Lacoste
(1
992),
Frachot and Lesne
(1
993), Heston
(1
99
I),
Pearson
and
Sun
(1994),
Pennachi
(1991),
and Stambaugh
(1988).
In such parametric special
cases, depending
on
the model specification and regularity conditions, one can generally
identify the parameters of
/A,
(5,
and
R,
to the extent that they affect bond prices, from
cross-sectional observations
of
the yield curve. For example, in the one-factor CIR model,
for which
r,
=
XI
evolves according to the stochastic differential equation
dX,
=
K(H
-
XI)
dt
+
yfi
d
W,,
one can identify
XI,
K,
0,
and
y
from essentially any four distinct bond
prices at time
T,
assuming
a
correct specification without measurement error. (Given the
likely misspecification of this model, this identification is not relied on in practice. Instead,
it
is common to use time series data and to assume fewer bond price observations at
a
given
time than parameters, or to assume measurement error, or both.) In order to estimate the
behavior of the state process
X
under the original probability measure
P,
one generally
must
resort to time-series observations,
so
as to capture the implied restrictions on the drift
process
u.
In this paper, although one
of
our goals is to classify
a
family of models that
is convenient for empirical work, we are
not
directly concerned with estimation issues.
We refer readers to the empirical studies cited above for such issues. We will restrict our
attention to behavior under one particular equivalent martingale measure
Q.
(There may
be
a
multiplicity of such measures in some cases, for example the case of jump diffusions
considered in Section
I
1
.)
As with all multifactor models, solving for all but
a
few types
of
derivative security
prices
is
computationally intensive. We present
a
practical finite-difference algorithm for
this purpose.
In
short,
we have a model specifiying simple relationships among yields and providing
term structure derivative prices, that is both computationally tractable and consistent with the
absence of arbitrage. While we have
not
described an economy whose general equilibrium
’Vincent Lacoste developed this point
of
view at a lecture at the Newton Institute at Cambridge University.
in
June
1995.
Vasicek
(1995)
develops the same idea.
A
YIELD-FACTOR
MODEL
OF
INTEREST RATES
381
implies the behavior of the term structure appearing in our model, that is easily done along
the lines of Cox, Ingersoll, and Ross (1985a,b) or Heston (1991) and adds little to what we
offer.
In
the model of Heath, Jarrow, and Morton
(
1992),
as
placed in a Markovian setting by
Musiela (1994), the state variable is,
in
essence, the entire current yield curve.
As
such, any
initial yield curve is, under regularity, consistent with the HJM model. Being in a finite-
dimensional state-space setting, our model has the disadvantage that not every initial yield
curve is consistent with a given paramaterization of the model.
(In
industry practice, this is
often handled by “calibration,” meaning the addition of time dependence to the coefficients
of the model in such a way as to match the given initial yield curve. That procedure has
obvious disadvantages.) The disadvantage of the finite-dimensional state-space setting can
also be one of its merits, for example in terms of numerical tractability.
In
any case, our
approach of taking yields
as
affine factors was independently accomplished within the HJM
setting by El Karoui and Lacoste (1992), taking the special Gaussian (constant volatility)
case. Their work has since been extended
to
the stochastic volatility case by Frachot, Janci,
and Lacoste
(I
992).
Other multifactor term structure models include those of Litterman and Scheinkman
(1
988), El Karoui, Myneni, and Viswanathan
(
19921, Jamshidian
(1
993), Chan
(I
992), and
Rogers (1995).
In
these models one could treat an unobserved factor
as
a “latent” variable
that can be filtered or otherwise calibrated from observations on the yield curve.
The remainder
of
the paper is organized
as
follows. Section
2
discusses the general
concepts involved in Markovian models of the yield curve. Section
3
specializes to a class
of “affine factor models,” in which yields are aftine in some abstract state variables. It is
shown that the yields are affine if, and essentially only if, the drift and diffusion functions of
the stochastic differential equation for
the
factors are also affine. Section
4
gives conditions
for existence and uniqueness of solutions to the associated stochastic differential equation.
Section
5
specializes to the case in which the factors are yields at fixed maturities. Sections
6
and
7
present examples of constant and stochastic volatility versions of the yield-factor
model, respectively, in which one of the factors is, for simplicity, the short rate itself.
Section
8
deals with the PDE for derivative prices, providing
a
change of variables that
orthogonalizes the diffusion
so
as to simplify the finite-difference solution. Sections 9 and
10
present examples
of
the solution to this PDE
in
the stochastic and deterministic volatility
cases, respectively, showing “convergence” to the theoretical solutions. Section
1
1
sketches
an extension to the case of jump diffusions.
2.
GENERAL FACTOR
MODELS
OF THE TERM STRUCTURE
For purposes of setting up the parametric model that we have in mind, we begin with the
general idea of a factor model for the yield curve. Under
a
given complete probability space
(Q,
F,
P)
and the augmented filtration’
[TI:
t
p
O}
generated by
a
standard Brownian
motion
W*
in
R”,
we suppose that there is a time-homogeneous Markov process
X
valued in
some open subset
D
of
R”
such that, for any times
t
and
t,
the market value
P,,~
at time
t
of
a
zero-coupon bondmaturing at time
t+t
is given by
f(X,,
T),
where
f
E
C2-’(D
x
[0,
00)).
The short-rate process
r
is assumed to be defined by continuity, in that there is
a
measurable
‘See. for
example,
Protter
(1990)
for definition5
involving the
theory
of
stochastic processes.
382
DARRELL DUFFIE
AND
RUI
KAN
function
R:
D
+
JR
defined as the limit of yields as maturity goes to zero, or
As is well understood from Harrison and Kreps
(1979)
and Harrison and Pliska
(198
I),
as well as others to follow, such as Ansel and Stricker
(1991),
only technical regularity
is
required for the equivalence between the absence
of
arbitrage and the existence of an
equivalent martingale measure; that is,
a
probability measure
Q
equivalent
to
P
under which
the price process
of
any security is a Q-martingale after normalization at each time
t
by the
value exp(&I
R(X,,)
ds)
of
continual reinvestment of interest from one
unit
of account held
from time zero at the short rate.
Suppose that
X
satisfies a stochastic differential equation of the form
where
u:
D
+
R”
anda:
D
--+
R“””
are regular enough for (2.2)
to
have a unique (strong)
solution. Additional technical regularity implies that there is a standard Brownian motion
W
in
Iw”
under
Q
such that
(2.3)
dX,
=
p(X,)dt
+o(X,)dW,,
where
p:
D
-+
R”
is a function that can be calculated
in
terms of
u,
a, and
,f.
General equi-
librium models
of
this form
of
asset pricing behavior are given by Cox, Ingersoll, and
Ross
(198%) and Huang
(1987).
The models
in
these papers are actually finite-horizon models
with time-dependent coefficients, but can be extended to time-homogeneous models
in
an
infinite-horizon setting. Our work here could be extended
to
time-dependent coefficients
merely
by
notational changes and minor technical regularity. Such time dependency, by
“calibration,” is standard operating procedure in trading implementations of term structure
models. See, for example, Ho and Lee
(I
986)
or Black, Derman, and Toy
(1
990).
Here, we are interested
in
choices
for
(,f,
p,
cr)
that are
compatible,
in
the sense that we
indeed have
(2.4)
f(X,,
T
-
t)
=
E
[
exp
(-
lT
R(X,)d.s)
1
X,]
as.,
0
5
f
5
T
<
cy7,
where
E
denotes expectation under Q. Expression (2.4) is merely the definition of
Q
as an
equivalent martingale measure, applied to zero-coupon bonds.
Of
course, it is relatively easy to construct compatible
(,f,
F,
cr).
For example, let
p,
cr,
and
R
be defined arbitrarily
so
that (2.3) and the right-hand side
of (2.4) are well defined, and then let
,f
be defined by (2.4). This is the “usual” approach
in arbitrage-based term structure models, as
in
Dothan
(1978),
Vasicek
(1977),
Richard
(1978),
Black, Derman, and Toy (1990), and
Hull
and White
(l990),
among many other
such models
in
which
X
is
the short rate itself and
R
is the identity. For multivariate models,
we have such examples as
El
Karoui, Myneni, and Viswanathan, (1992), Jamshidinn
(I
993),
A YIELD-FACTOR MODEL
OF
INTEKEST RATES
38.3
Beaglehole and Tenney (1991), and Rogers
(1
9951,
in
which
X
is Gauss-Markov (constant
a,
affine
p)
and
R
is a linear-quadratic form. (By “affine”
p
we mean, as usual, that there is
a constant matrix
u
and a vector
b
such that
p(x)
=
ux
+
h.)
Constantinides (1992) gives a
general equilibrium (representative agent) parametric model that implies this sort of linear-
quadratic-Gaussian behavior for short rates. There are also similar general equilibrium
models, such as those of Cox, Ingersoll, and Ross (1985b), Heston (1991), Longstaff and
Schwartz
(I
992), Nielsen and Sai-Requejo (1 992), and others in which one quickly arrives
at an expression such as (2.4)
in
which we can write
R(x)
=
xi
xi,
where the component
processes
X(”,
X”),
.
.
.
,
X(“)
are univariate processes satisfying the CIR equation
for scalar coefficients
u;
,
bi
,
c,
,
and
d,
.
These latter models are a special case of the model
presented later
in
this paper.
In any case, given any candidate for the short-rate process
r
satisfying mild regularity,
it
is easy
to
support
r
in a general equilibrium model based
on
a representative agent with,
say, HARA utility and a consumption process constructed
in
terms of
r.
(See,
for
example,
Heston 1991 and Duffie
1996,
Exercise
10.3.)
The available equilibrium models provide
useful theoretical relationships between the term structure, preferences, technology, and
macrovariables such as consumption, but have yet to add much to the practical day-to-day
problems of pricing and managing the risk
of fixed-income
instruments. For
our
purposes we
will follow the lead of others mentioned earlier by beginning directly with some compatible
model
(,f,
p,
a).
We are particularly interested
in
a class of models that is likely to be
numerically and empirically tractable, and eventually in models in which the state vector
X,
can be treated as an observation on the term structure itself, such as intended in the first
model of this sort due to Brennan and Schwartz
(
1979),
in
which the proposed factors are
the short rate and the yield on a consol. (The yield
on
a consol is the reciprocal of its price.
If one computes the price of a consol
in
the Brennan-Schwartz model, there is no reason to
expect the result
to
be the reciprocal of their state variable
C.
which is labeled the “consol
rate” by Brennan and Schwartz
for
expositional reasons. See Duffie, Ma, and Yong 1995
for an analysis of this issue.)
3.
AFFINE FACTOR MODELS
We consider a class
of
compatible models
(f,
p,
n)
with
for which, by virtue
of
the maintained assumption that
,f
E
C’.’(D
x
[0,
co)),
we know
that
A
and
B
are
C’
functions
on
[0,
00).
This parametric class of models, which we call
exponenfial-ufine
in light of the affine relationship between yields and factors, is relatively
tractable and offers some empirical advantages. In explaining the model, we use the fact
that
if
an affine relationship of the form
(Y
+
fi
.
.r
=
0
holds for all
x
in some nonempty
open Euclidean set, then
a
=
O
and
,B
=
0.
We call this the “matching principle.”
Since
f(x,
0)
=
1
for all
x
in
D,
which is an open set, (3.1) and the matching principle