YIELD CURVE SHAPES AND THE ASYMPTOTIC SHORT
RATE DISTRIBUTION IN AFFINE ONE-FACTOR MODELS
MARTIN KELLER-RESSEL AND THOMAS STEINER
Abstract. We consider a model for interest rates, where the short rate is
given under the risk-neutral measure by a time-homogenous, one-dimensional
affine process in the sense of Duffie, Filipović, and Schachermayer. We show
that in such a model yield curves can only be normal, inverse or humped
(i.e. endowed with a single local maximum). Each case can be characterized
by simple conditions on the present short rate rt. We give conditions under
which the short rate process will converge to a limit distribution and describe
the risk-neutral limit distribution in terms of its cumulant generating function.
We apply our results to the Vasiček model, the CIR model, a CIR model with
added jumps and a model of Ornstein-Uhlenbeck type.
1. Introduction
We consider a model for the term structure of interest rates, where the short
rate (rt)t≥0 is given under the martingale measure by a one-dimensional conserva-
tive affine process in the sense of Duffie, Filipović, and Schachermayer [2003]. An
affine short rate process of this type will lead to an exponentially-affine structure
of zero-coupon bond prices and thus also to an affine term structure of yields and
forward rates.
We emphasize here that the definition of Duffie et al. [2003] is not limited to diffu-
sions, but also includes processes with jumps and even with jumps whose intensity
depends in an affine way on the state of the process itself. The class of models we
consider naturally includes the Vasiček model, the CIR model and variants of them
that are obtained by adding jumps, such as the JCIR-model of Brigo and Mercurio
[2006, Section 22.8]. Since they are the best-known, the two ‘classical’ models of
Vasiček and Cox-Ingersoll-Ross will serve as the starting point for our discussion of
yield curve shapes:
A common criticism of the (time-homogenous) CIR and the Vasiček model is that
they are not flexible enough to accommodate more complex shapes of yield curves,
such as curves with a dip (a local minimum), curves with a dip and a hump, or
Date: November 4, 2018.
2000 Mathematics Subject Classification. 60J25, 91B28.
Key words and phrases. affine process, term structure of interest rates, Ornstein-Uhlenbeck
process, yield curve.
Supported by the Austrian Science Fund (FWF) through project P18022 and the START
programm Y328.
Supported by the module M5 “Modelling of Fixed Income Markets” of the PRisMa Lab,
financed by Bank Austria and the Republic of Austria through the Christian Doppler Research
Association.
Both authors would like to thank Josef Teichmann for most valuable discussions and encour-
agement. We also thank various proof-readers at FAM for their comments.
http://arxiv.org/abs/0704.0567v2
2 MARTIN KELLER-RESSEL AND THOMAS STEINER
other shapes that are frequently observed in the markets. Often these shortcomings
are explained by ‘too few parameters’ in the model (cf. Carmona and Tehranchi
[2006, Section 2.3.5] or Brigo and Mercurio [2006, Section 3.2]). However if jumps
are added to the mentioned models, additional parameters (potentially infinitely
many) are introduced through the jump part, while the model still remains in the
scope of affine models. It is not clear per se what consequences the introduction of
jumps will have for the range of attainable yield curves, and this is one question we
intend to answer in this article.
Moreover, there seems to be some confusion about what shapes of yield curves
are actually attainable even in well-studied models like the CIR-model. While
most sources (including the original paper of Cox et al. [1985]) mention inverse,
normal and humped shapes, Carmona and Tehranchi [2006, Section 2.3.5] write
that ‘tweaking the parameters [of the CIR model] can produce yield curves with one
hump or one dip’, and Brigo and Mercurio [2006, Section 3.2] state that ‘some typ-
ical shapes, like that of an inverted yield curve, may not be reproduced by the [CIR
or Vasiček] model.’ In our main result, Theorem 3.9, we settle this question and
prove that in any time-homogenous, affine one-factor model the attainable yield
curves are either inverse, normal or humped. The proof will rely only on tools
of elementary analysis and on the characterization of affine processes through the
generalized Riccati equations of Duffie et al. [2003].
Another related problem is how the shape of the yield curve is determined by the
parameters of the model, and also how – when the parameters are fixed – the yield
curve is determined by the level of the current short rate. We show in Section 4.2
that also in this respect the CIR model has not been completely understood and
discuss a misconception that originates in [Cox et al., 1985] and is repeated for ex-
ample in [Rebonato, 1998].
In Section 3.3 we provide conditions under which an affine process converges
to a limit distribution. We also characterize the limit distribution in terms of its
cumulant generating function, extending results of Jurek and Vervaat [1983] and
Sato and Yamazato [1984] for OU-type processes to the class of affine processes.
These results can again be interpreted in the context of interest rates, where they
can be used to derive the risk-neutral asymptotic distribution of the short rate
(rt)t≥0 as t goes to infinity.
We conclude our article in Section 4 by applying the theoretical results to several
interest rate models, such as the Vasiček model, the CIR model, the JCIR model
and an Ornstein-Uhlenbeck-type model.
2. Preliminaries
In this section we collect some key results on affine processes from Duffie et al.
[2003]. In their article affine processes are defined on the (m+n)-dimensional state
space Rm
>0 × Rn, and we will try to simplify notation where this is possible in the
one-dimensional case. Results on affine processes with state space R>0 can also be
found in Filipović [2001].
Definition 2.1 (One-dimensional affine process). A time-homogenous Markov pro-
cess (rt)t≥0 with state space D = R>0 or R and its semi-group (Pt)t≥0 are called
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 3
affine, if the characteristic function of its transition kernel pt(x, .), given by
p̂t(x, u) =
euξ pt(x, dξ)
and defined (at least) on
{u ∈ C : Reu ≤ 0} if D = R>0 ,
{u ∈ C : Reu = 0} if D = R ,
is exponentially affine in x. That is, there exist C-valued functions φ(t, u) and
ψ(t, u), defined on R>0 × U , such that
(2.1) p̂t(x, u) = exp (φ(t, u) + xψ(t, u)) for all x ∈ D, (t, u) ∈ R>0 × U .
For subsequent results the following regularity condition for (rt)t≥0 will be
needed:
Definition 2.2. An affine process is called regular if it is stochastically continuous
and the right hand derivatives
∂+t φ(t, u)|t=0 and ∂+t ψ(t, u)|t=0
exist for all u ∈ U and are continuous at u = 0.
Definition 2.3. The parameters (a, α, b, β, c, γ,m, µ) are called admissible for a
process with state space R>0 if
a = 0,
α, b, c, γ ∈ R>0 ,
β ∈ R ,
m, µ are Lévy measures on (0,∞), where m satisfies
(0,∞)
(ξ ∧ 1)m(dξ) <∞ ,
and admissible for a process with state space R if
a, c ∈ R>0 ,
b, β ∈ R ,
m is a Lévy measure on R \ {0} ,
α = 0, γ = 0, µ ≡ 0 .
Moreover define the truncation functions
hF (ξ) =
0 if D = R>0
if D = R
and hR(ξ) =
if D = R>0
0 if D = R
and finally the functions F (u), R(u) for u ∈ C as
F (u) = au2 + bu− c+
D\{0}
euξ − 1− uhF (ξ)
m(dξ) ,(2.2)
R(u) = αu2 + βu− γ +
D\{0}
euξ − 1− uhR(ξ)
µ(dξ) .(2.3)
The next result is a one-dimensional version of the key result of Duffie et al.
[2003]:
4 MARTIN KELLER-RESSEL AND THOMAS STEINER
Theorem 2.4 (Duffie, Filipović, and Schachermayer, Theorem 2.7). Suppose (rt)t≥0
is a one-dimensional regular affine process. Then it is a Feller process. Let A be
its infinitesimal generator. Then C∞c (D) is a core of A, C2c (D) ⊆ D(A) and there
exist some admissible parameters (a, α, b, β, c, γ,m, µ) such that, for f ∈ C2c (D),
Af(x) = (a+ αx)f ′′(x) + (b + βx)f ′(x)− (c+ γx)f(x)+
D\{0}
(f(x+ ξ)− f(x)− f ′(x)hF (ξ)) m(dξ)+
D\{0}
(f(x+ ξ)− f(x) − f ′(x)hR(ξ)) µ(dξ) .(2.4)
Moreover φ(t, u) and ψ(t, u), defined by (2.1), solve the generalized Riccati equa-
tions
∂t φ(t, u) = F (ψ(t, u)) , φ(0, u) = 0 ,(2.5a)
∂t ψ(t, u) = R (ψ(t, u)) , ψ(0, u) = u .(2.5b)
Conversely let (a, α, b, β, c, γ, µ,m) be some admissible parameters. Then there ex-
ists a unique regular affine semigroup (Pt)t≥0 with infinitesimal generator (2.4),
and (2.1) holds with φ(t, u) and ψ(t, u) given by (2.5).
Closely related to affine processes is the notion of an Ornstein-Uhlenbeck (OU-
)type process. These processes are of some importance, since they usually offer
good analytic tractability and have been studied for longer than affine processes.
Following Sato [1999, Chapter 17] an OU-type process (Xt)t≥0 can be defined as
the solution of the Langevin SDE
dXt = −λXt dt+ dLt, λ ∈ R, X0 ∈ R,
where (Lt)t≥0 is a Lévy process, often called background driving Lévy process
(BDLP). In an equivalent definition, an OU-type process is a time-homogenous
Markov process, whose transition kernel pt(x, .) has the characteristic function
p̂t(x, u) = exp
F (e−λsu) ds+ xe−λtu
where F (u) is the characteristic exponent of (Lt)t≥0. From the last equation it is
immediately seen that every OU-type process is an affine process in the sense of
Definition 2.1. It is also seen that in the generalized Riccati equations (2.5) for an
OU-type process necessarily R(u) = −λu. Comparing this with (2.3) and Defini-
tion 2.3, it is seen that any regular affine process with state space R is a process of
OU-type. The reverse, however is not true, as there also exist OU-type processes
with state space R>0. We will give an example of such a process in Section 4.4.
Naturally we will not only be interested in the process (rt)t≥0 itself, but also in
its integral
rs ds and in quantities of the type
(2.6) Qt f(x) := E
rs ds
f(rt)
∣∣∣∣ r0 = x
where f is a bounded function on D. The next result is an application of the
Feynman-Kac formula for Feller semigroups (cf. Rogers and Williams [1994, Sec-
tion III.19]) and can be found in Duffie et al. [2003]. It relies on the positivity of
(rt)t≥0 and is therefore only applicable if D = R>0.
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 5
Proposition 2.5 (Duffie, Filipović, and Schachermayer, Proposition 11.1). Let (rt)t≥0
be a one-dimensional, regular affine process with state space R>0. Then the fam-
ily (Qt)t≥0 defined by (2.6) forms a regular, affine semigroup with infinitesimal
generator
Bf(x) = Af(x)− xf(x) for all f ∈ C2c (D) .
We will make extensive use of the convexity and continuous differentiability of
the functions F and R from Definition 2.3. These properties are established in this
Lemma:
Lemma 2.6. If c = γ = 0 then F, R as defined in Definition 2.3 have the following
properties:
(i) R(0) = 0 and F (0) = 0.
(ii) R(u) <∞ for all u ∈ (−∞, 0].
(iii) If F (u) < ∞ on (c1, c2) ⊆ R, then F is either strictly convex on (c1, c2) or
F (u) = bu for all u ∈ R. The same holds for R with b replaced by β.
(iv) If F (u) <∞ on (c1, c2) ⊆ R, then F is continuously differentiable on (c1, c2).
Also the one-sided derivatives at c1 and c2 are defined but may take the values
−∞ (at c1) and +∞ (at c2). The same holds for R.
Proof. Property (i) is obvious. If D = R then by Definition 2.3 R(u) = βu such
that (ii) follows immediately. If D = R>0 we use the estimate
(2.7) |euξ − 1− uhR(ξ)| ≤ |u|
O(ξ2) ∧ 1
for all u ∈ (−∞, 0] and ξ ∈ R>0, and (ii) follows from (2.3). For Property (iii)
note that by the Lévy-Khintchine formula there exists an infinitely divisible random
variableX , such that F is its cumulant generating function, i.e. F (u) = logE
for u ∈ (c1, c2). Choosing two distinct numbers u, v ∈ (c1, c2), we apply the Cauchy-
Schwarz inequality to
= logE
2 · e vX2
≤ log
E[euX ] · E[evX ] = F (u) + F (v)
which shows convexity of F . The inequality is strict unless there exists some c 6= 0
such that euX = cevX almost surely. This can only be the case if X is con-
stant a.s., in which case F is linear. The same argument applies to R. Property
(iv) follows from the convexity and from the fact that F and R are analytic on
{u ∈ C : Reu ∈ (c1, c2)} (cf. Lukacs [1960, Chapter 7]). �
3. Theoretical Results
We will now use the theory from the last section to calculate bond prices, yields
and other quantities in an interest rate model where the short rate follows a one-
dimensional regular affine process (rt)t≥0 under the martingale measure. Naturally
we will also make the assumption that (rt)t≥0 is conservative, i.e. that pt(x,D) = 1
for all (t, x) ⊆ R>0 ×D. This implies by Duffie et al. [2003, Proposition 9.1] that
c = γ = 0 in Definition 2.3. We will need some additional assumptions which are
summarized in the following condition:
Condition 3.1. The one-dimensional affine process (rt)t≥0 is assumed to be reg-
ular and conservative. In addition, if the process has state space R, such that by
6 MARTIN KELLER-RESSEL AND THOMAS STEINER
Definition 2.3 R(u) = βu, we require that
(3.1) F (u) <∞ for all u ∈
(1/β, 0] if β < 0 ,
(−∞, 0] else .
It will be seen that the condition on F is necessary to guarantee existence of bond
prices for all maturities in the term structure model. By Sato [1999, Theorem 25.17]
we get an equivalent formulation of Condition 3.1, if we replace F (u) < ∞ by∫
|ξ|>1
euξm(dξ) <∞. Next we define a quantity that will generalize the coefficient
of mean reversion from OU-type processes:
Definition 3.2 (quasi-mean-reversion). Given a one-dimensional conservative affine
process (rt)t≥0, define the quasi-mean-reversion λ as the positive solution of
(3.2) R(−1/λ) = 1 .
If there is no positive solution we set λ = 0.
Since R is by Lemma 2.6 a convex function satisfying R(0) = 0, it is easy to see
that (3.2) can have at most one solution and thus λ is well-defined. The name quasi-
mean-reversion is derived from the fact that if (rt)t≥0 is a process of OU-type with
positive mean-reversion, then R(u) = βu and the quasi-mean-reversion λ = −β
is exactly the coefficient of mean reversion of (rt)t≥0. When the process (rt)t≥0
satisfies Condition 3.1, it is seen that F must be defined at least on (−1/λ, 0].
We will encounter several times the condition that λ > 0. The next result gives an
equivalent formulation in terms of (α, β, µ):
Proposition 3.3. The quasi-mean reversion λ is strictly positive if and only if
α > 0,
D\{0}
hR(ξ)µ(dξ) = ∞, or β −
D\{0}
hR(ξ)µ(dξ) < 0.
Proof. First note that by Lemma 2.6 R(u) < ∞ for all u ∈ (−∞, 0]. Using the
estimate (2.7) and a dominated convergence argument it is seen from (2.3) that
= α(3.3)
R(u)− αu2
= β0 := β −
D\{0}
hR(ξ)µ(dξ) ,(3.4)
where β0 can also take the value −∞. Suppose now that α > 0. Then by (3.3) we
get limu→−∞R(u) = ∞. Since R(0) = 0 and R is continuous it follows that there
exists a λ > 0 such that R(−1/λ) = 1. Similarly if α = 0, but β0 < 0, it follows
from (3.4) that limu→−∞R(u) = ∞ and thus again that λ > 0.
Conversely, suppose that α = 0 and β0 ≥ 0. Then
R′(u) = lim
= β0 ≥ 0 .
By the convexity of R it follows that R′(u) ≥ 0 for all u ∈ (−∞, 0). Since R(0) = 0
this implies that R(u) ≤ 0 for all u ∈ (−∞, 0), and consequently that λ = 0. �
3.1. Bond Prices. We consider now the price P (t, t + x) of a zero-coupon bond
with time to maturity x, at time t, given by
P (t, t+ x) = E
∫ t+x
rs ds
)∣∣∣∣Ft
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 7
The affine structure of (rt)t≥0 carries over to the bond prices, and we get the
following result:
Proposition 3.4. Let the short rate be given by a one-dimensional affine process
(rt)t≥0 satisfying Condition 3.1.
Then the bond price P (t, t+ x) exists for all t, x ≥ 0 and is given by
(3.5) P (t, t+ x) = exp (A(x) + rtB(x))
where A and B solve the generalized Riccati equations
∂xA(x) = F (B(x)) A(0) = 0 ,(3.6a)
∂xB(x) = R (B(x)) − 1 B(0) = 0 .(3.6b)
Proof. If D = R>0 the assertion follows directly from Proposition 2.5 by noting
that P (t, t+ x) = Qx 1.
If D = R then, as discussed after Theorem 2.4, (rt)t≥0 is a process of OU-type and
R(u) has the simple structure R(u) = βu. By Sato [1999, (17.2) - (17.3)] we obtain
in this case directly that
(3.7) E
∫ t+x
rs ds
= exp
F (B(s)) ds + rtB(x)
with B(x) = (1 − eβx)/β if β 6= 0 and B(x) = −x when β = 0. As a function of
x ∈ R>0, B is continuously decreasing from 0 to 1/β if β < 0, and from 0 to −∞
if β ≥ 0. It is therefore seen that the integral on the right side of (3.7) is finite for
all x ∈ R>0 if and only if F satisfies (3.1), as required by Condition 3.1. �
Corollary 3.5. Let (rt)t≥0 satisfy Condition 3.1 and have quasi-mean-reversion
λ. Then the function B(x) from Proposition 3.4 is strictly decreasing and satisfies
B(x) = −1/λ .
Proof. The result follows from a qualitative analysis of the autonomous ODE (3.6b).
Let λ > 0. Since R(−1/λ)−1 = 0 the point x∗ := −1/λ is an critical point of (3.6b).
By the convexity of R and the fact that R(0) = 0 it follows that R′(x∗) < 0 such
that x∗ is asymptotically stable, i.e. solutions entering a small enough neighborhood
of x∗ must converge to x∗. Since R(x)− 1 < 0 for x ∈ (x∗, 0] and there is no other
critical point in (x∗, 0], we conclude that B(x) – the solution of (3.6b) starting at
0 – is strictly decreasing and converges to x∗.
If λ = 0 then there is no critical point in (−∞, 0] and R(x)−1 < 0 for x ∈ (−∞, 0].
It follows that B(x) is strictly decreasing and diverges to −∞. �
3.2. The Yield Curve and the Forward Rate Curve. The next results are the
central theoretical results of this article and describe the global shapes of attainable
yield curves in any affine one-factor term structure model.
Definition 3.6. The (zero-coupon) yield Y (rt, x) is given by Y (rt, 0) := rt and
(3.8) Y (rt, x) := −
logP (t, t+ x)
= −A(x)
for all x > 0 .
For rt fixed, we call the function Y (rt, .) the yield curve.
The (instantaneous) forward rate f(rt, x) is given by f(rt, 0) := rt and
(3.9) f(rt, x) := −∂x logP (t, t+ x) = −A′(x)− rtB′(x) for all x > 0 .
For rt fixed, we call the function f(rt, .) the forward rate curve.
8 MARTIN KELLER-RESSEL AND THOMAS STEINER
By l’Hospital’s rule and the generalized Riccati equations (3.6) it is seen that
both the yield and the forward rate curve are continuous at 0.
The first quantity associated to the yield curve that we consider, is the asymptotic
level basymp of the yield curve as x → ∞, also known as long-term yield, consol
yield or simply ‘long end’.
Theorem 3.7. Let the short rate process be given by a one-dimensional affine pro-
cess (rt)t≥0 satisfying Condition 3.1 with quasi-mean-reversion λ.
If λ > 0 then
basymp := lim
Y (rt, x) = lim
f(rt, x) = −F (−1/λ) .
If λ = 0 then
basymp = lim
−F (u) + rt (1−R(u)) .
Proof. From (3.6a) we obtain that
(3.10) lim
= lim
A′(x) = lim
F (B(x)) .
If λ > 0 then by Corollary 3.5
(3.11) lim
B(x) = −1/λ, lim
= 0 and lim
B′(x) = 0
and the assertion follows by combining (3.8) – (3.11).
If λ = 0 then limx→∞B(x) = −∞ and
= lim
B′(x) = lim
R(B(x)) − 1 .
By setting u := B(x) we obtain the desired result. �
From Theorem 3.7 it is clear that for practical purposes only models with λ > 0
will be useful. So far we know that in this case the short end of the yield curve is
given by Y (rt, 0) = rt and the long end by Y (rt,∞) = basymp. We will now examine
what happens between these two endpoints.
Definition 3.8. The yield curve Y (rt, x) is called
• normal if it is a strictly increasing function of x,
• inverse if it is a strictly decreasing function of x,
• humped if it has exactly one local maximum and no minimum on (0,∞).
In addition we call the yield curve flat if it is constant over all x ∈ R>0.
This is our main result on the shapes of yield curves in affine one-factor models:
Theorem 3.9. Let the risk-neutral short rate process be given by a one-dimensional
affine process (rt)t≥0 satisfying Condition 3.1 and with quasi-mean-reversion λ > 0.
In addition suppose that F 6= 0 and that either F or R is non-linear. Then the
following holds:
• The yield curve Y (rt, .) can only be normal, inverse or humped.
• Define
bnorm := −
F ′(−1/λ)
R′(−1/λ) and binv :=
R′(0)
if R′(0) < 0
+∞ if R′(0) ≥ 0 .
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 9
The yield curve is normal if rt ≤ bnorm , humped if bnorm < rt < binv and
inverse if rt ≥ binv .
The above theorem is visualized in Figure 1. For its proof we will use a simple
Lemma. We state the Lemma without proof, since it follows in an elementary way
from the usual definition of a convex function on R.
Lemma 3.10. A strictly convex or a strictly concave function on R intersects an
affine function in at most two points. In the case of two intersection points p1 < p2,
the convex function lies strictly below the affine function on the interval (p1, p2); if
the function is concave it lies strictly above the affine function on (p1, p2).
Proof of Theorem 3.9. Define the function H(x) : R>0 → R by
(3.12) H(x) := Y (rt, x)x = −A(x)− rtB(x) .
We will see that the convexity behavior of H will be crucial for the shape of the
yield curve Y (rt, .). From the generalized Riccati equations (3.6) the first derivative
of H is calculated as
(3.13) ∂xH(x) = −F (B(x)) − rt (R(B(x))− 1)
and the second as
(3.14) ∂xxH(x) = −B′(x) (F ′(B(x)) + rtR′(B(x))) .
Note that F and R are continuously differentiable by Lemma 2.6, and also B by
(3.6b), such that the second derivative of H is well-defined and continuous. Since B
is strictly decreasing by Corollary 3.5, the factor −B′(x) is positive for all x ∈ R>0.
The sign of ∂xxH(x) therefore equals the sign of
(3.15) k(x) := F ′(B(x)) + rtR
′(B(x)) .
From the fact that B is decreasing and F and R are convex it is obvious that k
must be decreasing. We will now show that k has at most a single zero in [0,∞):
(a) D = R>0: We have assumed that either F or R is non-linear. By Lemma 2.6
this implies that either F or R is strictly convex, and thus that either F ′ or R′
is strictly increasing. If rt > 0, then it follows that k is strictly decreasing and
thus has at most a single zero. If rt = 0, an additional argument is needed:
It could happen that F is of the form F = bu such that k(x) = b and k is no
longer strictly decreasing. However, by assumption, F 6= 0 such that in this
case k has no zero in [0,∞).
(b) D = R: In this case, by the admissibility conditions in Definition 2.3, we
have necessarily R(u) = βu. Also, since either F or R is non-linear, F must
be non-linear and thus by Lemma 2.6 strictly convex. It follows that k(x) =
F ′(B(x))+rtβ is strictly decreasing and thus has at most a single zero in [0,∞).
We have shown that k is decreasing and has at most a single zero; to determine
whether it has a zero for some value of rt, we consider the two ‘endpoints’ k(0) and
limx→∞ k(x). First we show that
(3.16) k(0) ≥ 0 if and only if rt ≤ binv :=
R′(0)
if R′(0) < 0
+∞ if R′(0) ≥ 0 .
Since B(0) = 0 by Proposition 3.4 it follows that
k(0) = F ′(0) + rtR
′(0) .
10 MARTIN KELLER-RESSEL AND THOMAS STEINER
We distinguish two cases:
(a) If R′(0) < 0 then the assertion (3.16) follows immediately.
(b) Consider the case that R′(0) ≥ 0: Assume that D = R. Then we have
R(u) = βu and R′(0) = β ≥ 0. This, however, stands in contradiction to
our assumption λ > 0, which implies that β = −λ < 0 (cf. Definition 3.2).
Thus we must have D = R>0 and rt ≥ 0; in this case it follows that k(0) ≥ 0,
for all rt ∈ D, and we set binv = +∞.
Next we consider the right end of k(x) and show that
(3.17) lim
k(x) ≤ 0 if and only if rt ≥ bnorm := −
F ′(−1/λ)
R′(−1/λ) .
Since limx→∞B(x) = −1/λ by Corollary 3.5 we have that
(3.18) lim
k(x) = F ′(−1/λ) + rtR′(−1/λ) .
By assumption λ > 0, and by Definition 3.2 it holds that R(−1/λ) = 1. Also
R(0) = 0, and by the mean value theorem
1 = R(−1/λ)−R(0) = − 1
R′(ξ)
for some ξ ∈ (−1/λ, 0). Since R′ is increasing, it follows that R′(−1/λ) ≤ −λ < 0,
and we can deduce (3.17) directly from (3.18).
We summarize our results on the function k so far: k stays negative on (0,∞) if
rt ≥ binv and positive if rt ≤ bnorm. It has a single zero on (0,∞) if and only if
bnorm < rt < binv. If k has a zero on (0,∞), since k is decreasing, the sign of k will
be positive to the left of the zero and negative to the right of the zero.
Since ∂xxH has the same sign as k, the statements above translate in the obvious
way to the convexity behavior of H . We will now use the convexity behavior of H
to derive our results about the yield curve.
Consider the equation
(3.19) H(x) = cx, x ∈ [0,∞)
for some fixed c ∈ R. Since H(0) = 0 this equation has at least one solution,
x0 = 0. If rt ≥ binv then H(x) is strictly concave on [0,∞), and by Lemma 3.10
the equation (3.19) has at most one additional solution x1. Also, when the solution
exists, H(x) crosses cx from above at x1. Similarly if rt ≤ bnorm then H(x) is
strictly convex, and there exists at most one additional solution x2 to (3.19) on
[0,∞). If the solution exists, then cx is crossed from below at x2. In the last case
bnorm < rt < binv, there exists a x∗ – the zero of k(x) – such that H(x) is strictly
convex on (0, x∗) and strictly concave on (x∗,∞). Now there can exist at most two
additional solutions x1, x2 to (3.19) with x1 < x
∗ < x2, such that cx is crossed from
below at x1 and from above at x2.
Because of definition (3.12), every solution to (3.19), excluding x0 = 0, is also a
solution to
(3.20) Y (rt, x) = c, x ∈ (0,∞)
with rt fixed. Also the properties of crossing from above/below are preserved since
x is positive. This means that in the case rt ≥ binv, equation (3.20) has at most a
single solution, or in other words, that every horizontal line is crossed by the yield
curve at most in a single point. If it is crossed, it is crossed from above. This
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 11
implies that Y (x) is a strictly decreasing function of x, or following Definition 3.8,
that the yield curve is inverse. In the case rt ≤ bnorm we have again that (3.20) has
at most a single solution and that every horizontal line is crossed from below by
the yield curve, if it is crossed. In other words, the yield curve is normal. In the
last case of bnorm < rt < binv, the yield curve crosses every horizontal line at most
twice, in which case it crosses first from below, then from above. Thus in this case
the yield curve is humped. �
Corollary 3.11. Under the conditions of Theorem 3.9 the instantaneous forward
rate curve has the same global behavior as the yield curve, i.e.
Y (rt, .) is inverse ⇐⇒ f(rt, .) is strictly decreasing
Y (rt, .) is humped ⇐⇒
f(rt, .) has exactly one local maximum
and no local minimum
Y (rt, .) is normal ⇐⇒ f(rt, .) is strictly increasing .
In the second case the maximum of the forward rate curve is f(rt, x∗), where x∗
solves
(3.21) rt = −
F ′(B(x))
R′(B(x))
, x ∈ (0,∞) .
Proof. This follows from the fact that ∂xH(x) as given in (3.13) is exactly the
forward rate f(rt, x). The derivative of the forward rate is therefore ∂xxH(x),
which is given in (3.14) as
∂xf(rt, x) = ∂xxH(x) = −B′(x) · k(x) .
The factor −B′(x) 6= 0 is always positive, and the possible sign changes and zeroes
of k(x) are discussed in the proof of Theorem 3.9, leading to the stated equivalences.
Equation (3.21) is simply the condition k(x∗) = 0. �
Corollary 3.12. Under the conditions of Theorem 3.9 it holds that
(3.22) bnorm < basymp < binv
whenever the quantities are finite. In addition it holds that
(3.23) D ∩ (bnorm, binv) 6= ∅ .
Remark 3.13. Note that equation (3.23) implies that there is always some rt ∈ D
such that the yield curve Y (rt, .) is humped.
Proof. By the mean value theorem there exists a ξ ∈ (−1/λ, 0) such that
basymp = −F (−1/λ) = F (0)− F (−1/λ) =
F ′(ξ) .
Since F is convex and thus F ′ increasing, it holds that
(3.24)
F ′(−1/λ)
≤ basymp ≤
F ′(0)
Applying the mean value theorem to R, there exists another ξ ∈ (−1/λ, 0) such
1 = R(−1/λ)−R(0) = − 1
R′(ξ) .
12 MARTIN KELLER-RESSEL AND THOMAS STEINER
Since R′ is increasing we deduce that R′(−1/λ) ≤ −λ < 0. Assuming that also
R′(0) < 0 we get
(3.25) − 1
R′(−1/λ) ≤
≤ − 1
R′(0)
Since either F orR is non-linear, one of the functions is strictly convex by Lemma 2.6.
Consequently either both inequalities in (3.24) or in (3.25) are strict. Putting them
together we get
′(−1/λ)
R′(−1/λ) < basymp < −
F ′(0)
R′(0)
proving (3.22) under the assumption that R′(0) < 0.
If R′(0) ≥ 0 then by definition binv = ∞. Equation (3.24) still holds, but in
(3.25) only the left inequality sign remains valid. Together this still proves that
bnorm < basymp and we have shown (3.22).
To prove (3.23) we distinguish two cases:
(a) D = R. In this case it is sufficient to prove−∞ < binv and bnorm <∞. Consider
first binv. If R
′(0) ≥ 0 then by definition binv = ∞ and nothing is to prove.
If R′(0) < 0 then binv = −F ′(0)/R′(0). By convexity F ′(0) > −∞ and the
assertion follows. Consider now bnorm = −F ′(−1/λ)/R′(−1/λ). From (3.25)
we know that R′(−1/λ) ≤ −λ < 0. By convexity F ′(−1/λ) <∞ and it follows
that bnorm <∞.
(b) D = R>0. In this case it is sufficient to prove 0 ≤ bnorm and to apply (3.22).
As above we have that bnorm = −F ′(−1/λ)/R′(−1/λ) and that R′(−1/λ) ≤
−λ < 0. By Definition 2.3
F ′(−1/λ) = b+
(0,∞)
ξe−ξ/λm(dξ)
with b ≥ 0. It follows that F ′(−1/λ) ≥ 0, proving the assertion. �
The last Corollary of this section shows the interesting fact that the occurrence
of a humped yield curve is a necessary and sufficient sign of randomness in the
short rate model:
Corollary 3.14. Let the risk-neutral short rate process be given by a one-dimensional
affine process (rt)t≥0 satisfying Condition 3.1 with F 6= 0 and quasi-mean-reversion
λ > 0. Then the following statements are equivalent:
(i) There exists a rt ∈ D such that Y (rt, .) is flat.
(ii) There exists no rt ∈ D such that Y (rt, .) is humped.
(iii) The short rate process (rt)t≥0 is deterministic.
(iv) F (u) = bu and R(u) = βu.
Proof. Theorem 3.9, together with Corollary 3.12, shows already that ¬(iv) implies
¬(i) and ¬(ii). Also, from the form of the generator in (2.4), it is seen that (iii) and
(iv) are equivalent. It remains to show that (iv) implies (i) and (ii). Proceeding
as in the proof of Theorem 3.9 we obtain instead of (3.15) simply
k(x) = b+ rtβ .
The yield curve will be humped if and only if k has a single (isolated) zero in [0,∞).
Since k is a constant function, this cannot be the case for any rt ∈ D and we have
shown (ii). By the same arguments as in the proof of Theorem 3.9 the yield curve
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 13
binv = −
F’(0)
R’(0)
basymp = − F(− 1 λ)
bnorm = −
F’(− 1 λ)
R’(− 1 λ)
Time to Maturity
Figure 1. This Figure shows a graphical summary of Theorems
3.7 and 3.9, as well as the definitions of the key quantities bnorm,
basymp and binv. In any affine model satisfying the conditions of
Theorem 3.9, the shapes of yield curves will follow the picture
given here. They will be normal if r0 is below bnorm, humped if r0
is between bnorm and binv and inverse if r0 is above binv. Also all
yield curves will tend asymptotically to the same level basymp.
is flat if and only if k is constant and equal to 0. This is the case if rt = − bβ . It
remains to show that rt ∈ D. Note that β = −λ < 0. In particular β 6= 0, such
that for D = R we are already done. If D = R>0 we have by the admissibility
conditions in Definition 2.3 that b ≥ 0. Thus rt = − bβ ≥ 0 and we have shown
(i). �
3.3. The Limit Distribution of an Affine Process. It is well-known that the
Gaussian Ornstein-Uhlenbeck process, for example, converges in law to a limit
distribution and that this distribution is Gaussian. The goal of this section is to
establish a corresponding result for affine processes. While calculating the marginal
distributions of an affine process involves solving the generalized Riccati equations
(2.5), it will be seen that the limit distribution is much easier obtained and can be
determined directly from the functions F and R.
In the interest rate model considered in the preceding section, the short rate follows
an affine process under the martingale measure, such that the results will allow us
to characterize the risk-neutral asymptotic short rate distribution. Often also the
14 MARTIN KELLER-RESSEL AND THOMAS STEINER
limit distribution under the objective measure is of interest, but the affine prop-
erty is in general not preserved by an equivalent change of measure, such that the
results are not directly applicable. Nevertheless, for the sake of tractability, condi-
tions on the measure change can be imposed, such that the model is affine under
both the objective and the risk-neutral measure. (See Nicolato and Venardos [2003]
for an example from option pricing and Cheridito et al. [2005] for more general re-
sults). In such a setting the results can also be applied under the objective measure.
Before we state the result, we want to recall that a real-valued random variable
L is called self-decomposable if for every c ∈ (0, 1) there exists a random variable
Lc, independent of L, such that
L = cL+ Lc for all c ∈ (0, 1) .
Since self-decomposability is a distributional property, we will identify L and its
law, and refer to both as self-decomposable.
For OU-type processes, limit distributions have been studied for some time;
the first results can be found in Jurek and Vervaat [1983] and Sato and Yamazato
[1984]. The next theorem summarizes these results, and can be found in similar
form in Sato [1999, Theorem 17.5]:
Theorem 3.15. Let (rt)t≥0 be a OU-type process on R. If
β < 0 and
|ξ|>1
log |ξ|m(dξ) <∞
then (rt)t≥0 converges in law to a limit distribution L which is independent of r0
and has the following properties:
(i) L is self-decomposable.
(ii) The cumulant generating function κ(u) = log
eux dL(x) satisfies
(3.26) κ(iu) =
F (is)
ds for all u ∈ R .
Conversely, if L is a self-decomposable distribution on R and β < 0, there exists
a unique triplet (a, b,m) satisfying the admissibility conditions of Definition 2.3,
such that L is the limit distribution of the affine process (of OU-type) given by the
parameters (a, b,m, β).
As discussed in Section 2, every regular affine process with state space R is of
OU-type, such that the above theorem applies. We now state our corresponding
result for affine processes on R>0:
Theorem 3.16. Let (rt)t≥0 be a one-dimensional, regular, conservative affine pro-
cess with state space R>0. If
R′(0) < 0 and
log ξ m(dξ) <∞
then (rt)t≥0 converges in law to a limit distribution L which is independent of r0,
and whose cumulant generating function κ is given by
(3.27) κ(u) =
F (s)
ds for all u ∈ (−∞, 0] .
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 15
Proof. By Theorem 2.4 the transition kernel pt(x, .) of the process (rt)t≥0 has the
characteristic function
p̂t(x, u) = exp (φ(t, u) + xψ(t, u))
where φ and ψ satisfy the generalized Riccati equations (2.5) for all u ∈ U , and
thus in particular for all u ∈ (−∞, 0]. Since R(0) = 0, 0 is a critical point of the
autonomous ODE (2.5b), and by the assumption R′(0) < 0 it is asymptotically
stable. By the convexity of R, R′(0) < 0 also implies that R(u) > 0 for all u ∈
(−∞, 0), such that ψ(t, u) is a strictly increasing function in t for all u ∈ (−∞, 0).
Since 0 is the only critical point of (2.5b) on (−∞, 0] it also follows that
ψ(t, u) = 0 for all u ∈ (−∞, 0] .
Consequently,
(3.28) lim
log p̂t(x, u) = lim
φ(t, u) =
F (ψ(r, u)) dr =
F (s)
where the last two equalities follow from (2.5) and the transformation s = ψ(r, u).
We will now show that the last integral in (3.28) converges absolutely for all u ∈
(−∞, 0]: Since R(u) ≥ 0 and F (u) ≤ 0 for all u ∈ (−∞, 0] we obtain
F (s)
∣∣∣∣ ds = −
F (s)
ds ≤ − 1
R′(0)
F (s)
ds, u ∈ (−∞, 0] ,
where the inequality follows from the fact that the convex function R is supported by
its tangent at 0. From the definition of F (u) in (2.2) it is clear that the convergence
of the last integral will depend only on the jump part of F , i.e. the integral converges
if and only if
(3.29)
(0,∞)
esξ − 1
m(dξ) ds <∞, for all u ∈ (−∞, 0].
Define M(u, ξ) =
esξ−1
ds. For a fixed u ∈ (−∞, 0], it is easily verified that
M(u, ξ) = O(ξ) as ξ → 0, and that M(u, ξ) = O(log ξ) as ξ → ∞. Since the Lévy
measurem(dξ) integrates (ξ∧1) by Definition 2.3, and log ξ ·1{ξ>1} by assumption,
it must also integrateM(u, ξ). Applying Fubini’s theorem, (3.29) follows, such that
κ(u) :=
F (s)
ds converges for all u ∈ (−∞, 0]. In particular limu↑0 κ(u) = 0, such
that the limit in (3.28) is a function that is left-continuous at 0. By standard results
on Laplace transforms of probability measures (cf. Steutel and van Harn [2004,
Theorem A.3.1]), the pointwise convergence of cumulant generating functions to a
function that is left-continuous at 0 implies convergence in distribution of (rt)t≥0
to a limit distribution L with cumulant generating function given by (3.28). �
Since the marginal distributions of an affine process are infinitely divisible, also
the limit distribution L must be infinitely divisible, if it exists. In Theorem 3.15
a stronger result is given for an affine process on R: In this case L is also self-
decomposable. An obvious question is, if this result can be extended to the state
space R>0. We will see that the answer is negative. In Section 4.3 an example of an
affine process with state space R>0 is given, which converges to an infinitely divisible
limit distribution that is not self-decomposable. This result is interesting, since it
leaves open the possibility of some unexpected properties of the limit distribution of
16 MARTIN KELLER-RESSEL AND THOMAS STEINER
an affine process. For example a self-decomposable distribution is always unimodal,
whereas an infinitely divisible distribution might be not.
4. Applications
4.1. The Vasiček model. We apply the results of the last section to the classical
Vasiček model
(4.1) drt = −λ(rt − θ) dt+ σ dWt, r0 ∈ R
where (Wt)t≥0 is a standard Brownian motion under the risk-neutral measure and
λ, θ, σ > 0. The Vasiček model is arguably the simplest affine model, and no
surprises are to be expected here. In fact all results that we state here can already
be found in the original paper of Vasiček [1977]. We advise the reader to view this
paragraph as a warm-up for the following examples.
Clearly (rt)t≥0 is a conservative affine process with
F (u) = λθu +
u2 ,(4.2)
R(u) = −λu .(4.3)
From the quadratic term in F and Definition 2.3, it is seen that (rt)t≥0 has state
space R. This property is often criticized, since it allows the short rate to become
negative.
From Theorem 3.9 we calculate
binv = θ and bnorm = θ −
such that the yield curve in the Vasiček model is normal if rt ≤ θ − σ2/λ2, inverse
if rt ≥ θ and humped in the remaining cases.
The long term yield is calculated from (3.7) as
basymp = −F (−1/λ) = θ −
in this case exactly the arithmetic mean of binv and bnorm.
Theorem 3.15 applies and the cumulant generating function κ of the risk-neutral
limit distribution L satisfies
κ(iu) = − 1
F (is)
iθ − σ
ds = uiθ − u
for u ∈ R. Hence, L is Gaussian with mean θ and variance σ2
4.2. The Cox-Ingersoll-Ross model. The Cox-Ingersoll-Ross (CIR)-model was
introduced by Cox et al. [1985]. In this model the short rate process (rt)t≥0 is given
by the SDE
(4.4) drt = −a(rt − θ)dt+ σ
rt dWt, r0 ∈ R>0
where (Wt)t≥0 is a standard Brownian Motion under the risk-neutral measure and
a, θ, σ > 0. The process (rt)t≥0 is a conservative affine process with
F (u) = aθu ,(4.5)
R(u) =
u2 − au .(4.6)
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 17
From Definition 2.3 it is seen that (rt)t≥0 has state space R>0. The fact that
interest rates stay non-negative in the CIR-model is often cited as an advantage
of the model over the Vasiček model. Calculating the quasi-mean-reversion (see
Definition 3.2), we find that
a2 + 2σ2 + a
From Theorem 3.7 we find that the long-term yield is given by
basymp = −F (−1/λ) =
a2 + 2σ2 + a
The boundary between humped and inverse behavior binv is calculated from Theo-
rem 3.9 as
binv = −
F ′(0)
R′(0)
Both quantities basymp and binv can also be found in [Cox et al., 1985, Eq. (26) and
following paragraph]. Before we consider bnorm, we quote (with notation adapted
to (4.4)) from page 394 of [Cox et al., 1985] where the shape of the yield curve is
discussed:
‘When the spot rate is below the long-term yield [= basymp], the
term structure is uniformly rising. With an interest rate in excess
of θ [= binv], the term structure is falling. For intermediate values
of the interest rate, the yield curve is humped.’
In our terminology, they claim that the yield curve is normal for rt ≤ basymp,
humped for basymp < rt < binv and inverse for rt ≥ binv. This stands in clear
contradiction to Theorem 3.9 and Corollary 3.12 where we have obtained that yield
curves are normal if and only if rt ≤ bnorm and that bnorm < basymp, or – in plain
words – that there are yield curves starting strictly below the long-term yield that
are still humped.
The claims of Cox et al. [1985] are repeated in [Rebonato, 1998, p. 244f], where even
several plots of ‘yield surfaces’ (the yield as a function of rt and x) are presented as
evidence. However Rebonato fails to indicate the level of basymp in the plots, such
that the conclusion remains ambiguous.
To clarify the scope of humped yield curves in the CIR-model we calculate bnorm
from Theorem 3.9:
bnorm = −
F ′(−1/λ)
R′(−1/λ) =
a2 + 2σ2
The relation bnorm < basymp < binv is immediately confirmed by noting that basymp
is the harmonic mean of bnorm and binv. For a graphical illustration we refer to
the second yield curve from below in Figure 1. The plot actually shows CIR yield
curves with parameters
a = 0.5, σ = 0.5, θ = 6%
plotted over a time scale of 25 years. The second curve from below starts at
r0 = 4.2%, i.e. below the long-term yield, but is visibly humped.
18 MARTIN KELLER-RESSEL AND THOMAS STEINER
To calculate the limit distribution of (rt)t≥0, we apply Theorem 3.16: The cu-
mulant generating function κ(u) of the limit distribution is given by
κ(u) =
F (s)
1− sσ2/2a ds = −
This is the cumulant generating function of a gamma distribution with shape pa-
rameter 2aθ/σ2 and scale parameter σ2/2a. Again this result can already be found
in Cox et al. [1985, p. 392].
4.3. An extension of the CIR model. To illustrate the power of the affine
setting, we consider now an extension of the CIR model that is obtained by adding
jumps to (4.4). We define the risk-neutral short rate process by
(4.7) drt = −a(rt − θ)dt+ σ
rt dWt + dJt, r0 ≥ 0
where (Jt)t≥0 is a compound Poisson process with intensity c > 0 and expo-
nentially distributed jumps of mean ν > 0. This model has been introduced by
Duffie and Gârleanu [2001] as a model for default intensity and is used by Filipović
[2001] as a short rate model. It can also be found in Brigo and Mercurio [2006]
under the name JCIR model. It is easily calculated that
F (u) = aθu+
ν − u, u ∈ (−∞, ν) ,(4.8)
R(u) =
u2 − au .(4.9)
Solving the generalized Riccati equations (3.6) for A(x) and B(x) becomes quite
tedious, but the quantities binv, basymp, bnorm can be calculated from Theorem 3.7
and Theorem 3.9 in a few lines: The quasi-mean reversion λ stays the same as in
the CIR model, since R does not change. From
F ′(u) = aθ +
(ν − u)2
we derive immediately
binv = θ +
basymp =
ν(a+ ν) + 2
bnorm =
γ(σ2ν + γ − a)2 ,
where γ =
a2 + 2σ2. Note that by setting the jump intensity c to zero, the ex-
pressions of the (original) CIR model are recovered.
Next we calculate the limit distribution of the model. Using the abbreviations
ρ := σ2/2 and ∆ := a− νρ we obtain
κ(u) =
F (s)
1− sρ/a ds+ c
(s− ν)(ρs− a) =
if ∆ 6= 0
−θν log
if ∆ = 0
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 19
as the cumulant generating function of the limit distribution L under the martin-
gale measure.
We now take a closer look at the distribution L, since this will answer the
question raised at the end of Section 3.3: For certain parameters, L is an example
for a limit distribution of an affine process that is infinitely divisible, but not self-
decomposable. We consider the case ∆ = 0 and define
(4.10) l(x) :=
νe−νx, x ∈ R>0 .
By Frullani’s integral formula
(4.11) κ(u) =
(eux − 1) l(x)
for all u ∈ (−∞, ν). Since l is non-negative on R>0, l(x)/x is the density of a
Lévy measure and (4.11) is seen to be the Lévy-Khintchine representation for the
cumulant generating function of the infinitely divisible distribution L. In addition,
L is self-decomposable if and only if l is non-negative and non-increasing on R>0
(cf. Sato [1999, Corollary 15.11]).
In the case of l(x) given by (4.10), it is easily calculated that l(x) has a single
maximum at x∗ = 1
. Thus, if c ≤ aθν, then x∗ ≤ 0, such that l is non-
increasing on R>0 and L is self-decomposable. If c > aθν then l is increasing
in the interval [0, x∗) and the limit distribution L is infinitely divisible, but not
self-decomposable.
4.4. The gamma model. Instead of analyzing the properties of a known model,
we will now follow a different route and construct a model that satisfies some given
properties. We want to construct an affine process on R>0 that has the same limit
distribution as the CIR model (i.e. a gamma distribution), but is a process of OU-
type. The second property is equivalent to R(u) = βu. Considering Theorem 3.16,
we know that if we want to obtain a limit distribution, we need β < 0. To keep
with the notation of the Vasiček model, we will write R(u) = −λu where λ > 0.
Now by (3.27) the cumulant generating function of the limit distribution is given
(4.12) κ(u) =
F (s)
ds for all u ∈ (−∞, 0] .
Let the limit distribution be a gamma distribution with shape parameter k > 0
and scale parameter θ > 0. Then κ(u) = −k log(1 − θu) and by (4.12)
F (u) =
1− θu .
Setting c = λk and ν = 1/θ it is seen that F (u) is equal to the last term in
(4.8). This means that the driving Lévy process of (rt)t≥0 is of the same kind
as the process (Jt)t≥0 in (4.7), i.e. (rt)t≥0 is a pure jump OU-type process with
exponentially distributed jump heights of mean 1/θ and with jump intensity λk.
We interpret the affine process we have constructed as a risk-neutral short rate
process. It is clear that the bond prices are of the exponentially-affine form (3.5).
From the generalized Riccati equation (3.6b) we obtain
B(x) =
e−λx − 1
20 MARTIN KELLER-RESSEL AND THOMAS STEINER
From equation (3.6a) we calculate
A(x) =
F (B(s)) ds =
θ + λ
(log(1− θB(x)) − θx) ,
such that the bond prices are given by
P (t, t+ x) = exp
−x λθk
θ + λ
+ rtB(x)
(1− θB(x))
θ+λ .
The global shape of the yield curve is described by the quantities
binv = kθ, basymp =
1/θ+ 1/λ
, bnorm =
(1/θ + 1/λ)2
and it is seen that for the gamma-OU-process basymp is the geometric average of
binv and bnorm.
5. Conclusions
In this article we have given, under very general conditions, a characterization of
the yield curve shapes that are attainable in term structure models where the risk-
neutral short rate is given by a time-homogenous, one-dimensional affine process.
Even though the parameter space for this class of models is infinite-dimensional,
the scope of attainable yield curves is very narrow, with only three possible global
shapes. In addition we have given conditions under which an affine process con-
verges to a limit distribution, and we have characterized the limit distribution in
terms of its cumulant generating function, extending some known results on OU-
type processes.
The most obvious question for future research is the extension of these results to
multi-factor models. It is evident from numerical results that in two-factor models
yield curves with e.g. a dip, or also with a dip and a hump, can be obtained. It
would be interesting to see if more complex shapes can also be produced, or if there
are similar limitations as in the single-factor case. Also, in the one-factor case the
dependence of the yield curve shape on the current short rate is basically described
by the intervals D ∩ (−∞, bnorm], (bnorm, binv) and [binv,∞). In the two-factor case
the partitioning of the state-space might be more complex, and we expect to see
more interesting transitions between yield curve types. Another aspect is, that
since affine processes as a general framework become better understood, extensions
of classical models e.g. by adding jumps, like in the JCIR model described in
Section 4.3, become more feasible and attractive for applications.
References
Damiano Brigo and Fabio Mercurio. Interest Rate Models - Theory and Practice.
Springer Finance. Springer, 2nd edition, 2006.
René Carmona and Michael Tehranchi. Interest Rate Models: An Infinite Dimen-
sional Stochastic Analysis Perspective. Springer Finance. Springer, 2006.
Patrick Cheridito, Damir Filipović, and Marc Yor. Equivalent and absolutely con-
tinuous measure changes for jump-diffusion processes. The Annals of Applied
Probability, 15(3), 2005.
John C. Cox, Jonathan E. Ingersoll, and Stephen A. Ross. A theory on the term
structure of interest rates. Econometrica, 53(2):385–407, 1985.
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 21
Darrell Duffie and Nicolae Gârleanu. Risk and valuation of collateralized debt
obligations. Financial Analysts Journal, 57(1):41 – 59, 2001.
Darrell Duffie, Damir Filipović, and Walter Schachermayer. Affine processes and
applications in finance. The Annals of Applied Probability, 13(3):984–1053, 2003.
Damir Filipović. A general characterization of one factor affine term structure
models. Finance and Stochastics, 5:389–412, 2001.
Zbigniew J. Jurek and Wim Vervaat. An integral representation for self-
decomposable Banach space valued random variables. Zeitschrift für Wahrschein-
lichkeitstheorie und verwandte Gebiete, 62:247–262, 1983.
Eugene Lukacs. Characteristic Functions. Charles Griffin & Co Ltd., 1960.
Elisa Nicolato and Emmanouil Venardos. Option pricing in stochastic volatility
models of the Ornstein-Uhlenbeck type. Mathematical Finance, 13 (4):445–466,
2003.
Riccardo Rebonato. Interest-Rate Option Models. Wiley, 2nd edition, 1998.
L.C.G. Rogers and David Williams. Diffusions, Markov Processes and Martingales,
Volume 1. Cambridge Mathematical Library, 2nd edition, 1994.
Ken-iti Sato. Lévy processes and infinitely divisible distributions. Cambridge Uni-
versity Press, 1999.
Ken-iti Sato and M. Yamazato. Operator-selfdecomposable distributions as limit
distributions of processes of Ornstein-Uhlenbeck type. Stochastic Processes and
Applications, 17:73–100, 1984.
Fred Steutel and Klaas van Harn. Infinite Divisibility of Probability Distributions
on the Real Line. Marcel Dekker Inc., 2004.
Oldrich Vasiček. An equilibrium characterization of the term structure. Journal of
Financial Economics, 5:177–188, 1977.
Vienna University of Technology, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria
E-mail address: mkeller@fam.tuwien.ac.at
Vienna University of Technology, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria
E-mail address: thomas@fam.tuwien.ac.at
	1. Introduction
	2. Preliminaries
	3. Theoretical Results
	3.1. Bond Prices
	3.2. The Yield Curve and the Forward Rate Curve
	3.3. The Limit Distribution of an Affine Process
	4. Applications
	4.1. The Vasicek model
	4.2. The Cox-Ingersoll-Ross model
	4.3. An extension of the CIR model
	4.4. The gamma model
	5. Conclusions
	References

  We consider a model for interest rates, where the short rate is given by a
time-homogenous, one-dimensional affine process in the sense of Duffie,
Filipovic and Schachermayer. We show that in such a model yield curves can only
be normal, inverse or humped (i.e. endowed with a single local maximum). Each
case can be characterized by simple conditions on the present short rate. We
give conditions under which the short rate process will converge to a limit
distribution and describe the limit distribution in terms of its cumulant
generating function. We apply our results to the Vasicek model, the CIR model,
a CIR model with added jumps and a model of Ornstein-Uhlenbeck type.

Introduction
We consider a model for the term structure of interest rates, where the short
rate (rt)t≥0 is given under the martingale measure by a one-dimensional conserva-
tive affine process in the sense of Duffie, Filipović, and Schachermayer [2003]. An
affine short rate process of this type will lead to an exponentially-affine structure
of zero-coupon bond prices and thus also to an affine term structure of yields and
forward rates.
We emphasize here that the definition of Duffie et al. [2003] is not limited to diffu-
sions, but also includes processes with jumps and even with jumps whose intensity
depends in an affine way on the state of the process itself. The class of models we
consider naturally includes the Vasiček model, the CIR model and variants of them
that are obtained by adding jumps, such as the JCIR-model of Brigo and Mercurio
[2006, Section 22.8]. Since they are the best-known, the two ‘classical’ models of
Vasiček and Cox-Ingersoll-Ross will serve as the starting point for our discussion of
yield curve shapes:
A common criticism of the (time-homogenous) CIR and the Vasiček model is that
they are not flexible enough to accommodate more complex shapes of yield curves,
such as curves with a dip (a local minimum), curves with a dip and a hump, or
Date: November 4, 2018.
2000 Mathematics Subject Classification. 60J25, 91B28.
Key words and phrases. affine process, term structure of interest rates, Ornstein-Uhlenbeck
process, yield curve.
Supported by the Austrian Science Fund (FWF) through project P18022 and the START
programm Y328.
Supported by the module M5 “Modelling of Fixed Income Markets” of the PRisMa Lab,
financed by Bank Austria and the Republic of Austria through the Christian Doppler Research
Association.
Both authors would like to thank Josef Teichmann for most valuable discussions and encour-
agement. We also thank various proof-readers at FAM for their comments.
http://arxiv.org/abs/0704.0567v2
2 MARTIN KELLER-RESSEL AND THOMAS STEINER
other shapes that are frequently observed in the markets. Often these shortcomings
are explained by ‘too few parameters’ in the model (cf. Carmona and Tehranchi
[2006, Section 2.3.5] or Brigo and Mercurio [2006, Section 3.2]). However if jumps
are added to the mentioned models, additional parameters (potentially infinitely
many) are introduced through the jump part, while the model still remains in the
scope of affine models. It is not clear per se what consequences the introduction of
jumps will have for the range of attainable yield curves, and this is one question we
intend to answer in this article.
Moreover, there seems to be some confusion about what shapes of yield curves
are actually attainable even in well-studied models like the CIR-model. While
most sources (including the original paper of Cox et al. [1985]) mention inverse,
normal and humped shapes, Carmona and Tehranchi [2006, Section 2.3.5] write
that ‘tweaking the parameters [of the CIR model] can produce yield curves with one
hump or one dip’, and Brigo and Mercurio [2006, Section 3.2] state that ‘some typ-
ical shapes, like that of an inverted yield curve, may not be reproduced by the [CIR
or Vasiček] model.’ In our main result, Theorem 3.9, we settle this question and
prove that in any time-homogenous, affine one-factor model the attainable yield
curves are either inverse, normal or humped. The proof will rely only on tools
of elementary analysis and on the characterization of affine processes through the
generalized Riccati equations of Duffie et al. [2003].
Another related problem is how the shape of the yield curve is determined by the
parameters of the model, and also how – when the parameters are fixed – the yield
curve is determined by the level of the current short rate. We show in Section 4.2
that also in this respect the CIR model has not been completely understood and
discuss a misconception that originates in [Cox et al., 1985] and is repeated for ex-
ample in [Rebonato, 1998].
In Section 3.3 we provide conditions under which an affine process converges
to a limit distribution. We also characterize the limit distribution in terms of its
cumulant generating function, extending results of Jurek and Vervaat [1983] and
Sato and Yamazato [1984] for OU-type processes to the class of affine processes.
These results can again be interpreted in the context of interest rates, where they
can be used to derive the risk-neutral asymptotic distribution of the short rate
(rt)t≥0 as t goes to infinity.
We conclude our article in Section 4 by applying the theoretical results to several
interest rate models, such as the Vasiček model, the CIR model, the JCIR model
and an Ornstein-Uhlenbeck-type model.
2. Preliminaries
In this section we collect some key results on affine processes from Duffie et al.
[2003]. In their article affine processes are defined on the (m+n)-dimensional state
space Rm
>0 × Rn, and we will try to simplify notation where this is possible in the
one-dimensional case. Results on affine processes with state space R>0 can also be
found in Filipović [2001].
Definition 2.1 (One-dimensional affine process). A time-homogenous Markov pro-
cess (rt)t≥0 with state space D = R>0 or R and its semi-group (Pt)t≥0 are called
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 3
affine, if the characteristic function of its transition kernel pt(x, .), given by
p̂t(x, u) =
euξ pt(x, dξ)
and defined (at least) on
{u ∈ C : Reu ≤ 0} if D = R>0 ,
{u ∈ C : Reu = 0} if D = R ,
is exponentially affine in x. That is, there exist C-valued functions φ(t, u) and
ψ(t, u), defined on R>0 × U , such that
(2.1) p̂t(x, u) = exp (φ(t, u) + xψ(t, u)) for all x ∈ D, (t, u) ∈ R>0 × U .
For subsequent results the following regularity condition for (rt)t≥0 will be
needed:
Definition 2.2. An affine process is called regular if it is stochastically continuous
and the right hand derivatives
∂+t φ(t, u)|t=0 and ∂+t ψ(t, u)|t=0
exist for all u ∈ U and are continuous at u = 0.
Definition 2.3. The parameters (a, α, b, β, c, γ,m, µ) are called admissible for a
process with state space R>0 if
a = 0,
α, b, c, γ ∈ R>0 ,
β ∈ R ,
m, µ are Lévy measures on (0,∞), where m satisfies
(0,∞)
(ξ ∧ 1)m(dξ) <∞ ,
and admissible for a process with state space R if
a, c ∈ R>0 ,
b, β ∈ R ,
m is a Lévy measure on R \ {0} ,
α = 0, γ = 0, µ ≡ 0 .
Moreover define the truncation functions
hF (ξ) =
0 if D = R>0
if D = R
and hR(ξ) =
if D = R>0
0 if D = R
and finally the functions F (u), R(u) for u ∈ C as
F (u) = au2 + bu− c+
D\{0}
euξ − 1− uhF (ξ)
m(dξ) ,(2.2)
R(u) = αu2 + βu− γ +
D\{0}
euξ − 1− uhR(ξ)
µ(dξ) .(2.3)
The next result is a one-dimensional version of the key result of Duffie et al.
[2003]:
4 MARTIN KELLER-RESSEL AND THOMAS STEINER
Theorem 2.4 (Duffie, Filipović, and Schachermayer, Theorem 2.7). Suppose (rt)t≥0
is a one-dimensional regular affine process. Then it is a Feller process. Let A be
its infinitesimal generator. Then C∞c (D) is a core of A, C2c (D) ⊆ D(A) and there
exist some admissible parameters (a, α, b, β, c, γ,m, µ) such that, for f ∈ C2c (D),
Af(x) = (a+ αx)f ′′(x) + (b + βx)f ′(x)− (c+ γx)f(x)+
D\{0}
(f(x+ ξ)− f(x)− f ′(x)hF (ξ)) m(dξ)+
D\{0}
(f(x+ ξ)− f(x) − f ′(x)hR(ξ)) µ(dξ) .(2.4)
Moreover φ(t, u) and ψ(t, u), defined by (2.1), solve the generalized Riccati equa-
tions
∂t φ(t, u) = F (ψ(t, u)) , φ(0, u) = 0 ,(2.5a)
∂t ψ(t, u) = R (ψ(t, u)) , ψ(0, u) = u .(2.5b)
Conversely let (a, α, b, β, c, γ, µ,m) be some admissible parameters. Then there ex-
ists a unique regular affine semigroup (Pt)t≥0 with infinitesimal generator (2.4),
and (2.1) holds with φ(t, u) and ψ(t, u) given by (2.5).
Closely related to affine processes is the notion of an Ornstein-Uhlenbeck (OU-
)type process. These processes are of some importance, since they usually offer
good analytic tractability and have been studied for longer than affine processes.
Following Sato [1999, Chapter 17] an OU-type process (Xt)t≥0 can be defined as
the solution of the Langevin SDE
dXt = −λXt dt+ dLt, λ ∈ R, X0 ∈ R,
where (Lt)t≥0 is a Lévy process, often called background driving Lévy process
(BDLP). In an equivalent definition, an OU-type process is a time-homogenous
Markov process, whose transition kernel pt(x, .) has the characteristic function
p̂t(x, u) = exp
F (e−λsu) ds+ xe−λtu
where F (u) is the characteristic exponent of (Lt)t≥0. From the last equation it is
immediately seen that every OU-type process is an affine process in the sense of
Definition 2.1. It is also seen that in the generalized Riccati equations (2.5) for an
OU-type process necessarily R(u) = −λu. Comparing this with (2.3) and Defini-
tion 2.3, it is seen that any regular affine process with state space R is a process of
OU-type. The reverse, however is not true, as there also exist OU-type processes
with state space R>0. We will give an example of such a process in Section 4.4.
Naturally we will not only be interested in the process (rt)t≥0 itself, but also in
its integral
rs ds and in quantities of the type
(2.6) Qt f(x) := E
rs ds
f(rt)
∣∣∣∣ r0 = x
where f is a bounded function on D. The next result is an application of the
Feynman-Kac formula for Feller semigroups (cf. Rogers and Williams [1994, Sec-
tion III.19]) and can be found in Duffie et al. [2003]. It relies on the positivity of
(rt)t≥0 and is therefore only applicable if D = R>0.
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 5
Proposition 2.5 (Duffie, Filipović, and Schachermayer, Proposition 11.1). Let (rt)t≥0
be a one-dimensional, regular affine process with state space R>0. Then the fam-
ily (Qt)t≥0 defined by (2.6) forms a regular, affine semigroup with infinitesimal
generator
Bf(x) = Af(x)− xf(x) for all f ∈ C2c (D) .
We will make extensive use of the convexity and continuous differentiability of
the functions F and R from Definition 2.3. These properties are established in this
Lemma:
Lemma 2.6. If c = γ = 0 then F, R as defined in Definition 2.3 have the following
properties:
(i) R(0) = 0 and F (0) = 0.
(ii) R(u) <∞ for all u ∈ (−∞, 0].
(iii) If F (u) < ∞ on (c1, c2) ⊆ R, then F is either strictly convex on (c1, c2) or
F (u) = bu for all u ∈ R. The same holds for R with b replaced by β.
(iv) If F (u) <∞ on (c1, c2) ⊆ R, then F is continuously differentiable on (c1, c2).
Also the one-sided derivatives at c1 and c2 are defined but may take the values
−∞ (at c1) and +∞ (at c2). The same holds for R.
Proof. Property (i) is obvious. If D = R then by Definition 2.3 R(u) = βu such
that (ii) follows immediately. If D = R>0 we use the estimate
(2.7) |euξ − 1− uhR(ξ)| ≤ |u|
O(ξ2) ∧ 1
for all u ∈ (−∞, 0] and ξ ∈ R>0, and (ii) follows from (2.3). For Property (iii)
note that by the Lévy-Khintchine formula there exists an infinitely divisible random
variableX , such that F is its cumulant generating function, i.e. F (u) = logE
for u ∈ (c1, c2). Choosing two distinct numbers u, v ∈ (c1, c2), we apply the Cauchy-
Schwarz inequality to
= logE
2 · e vX2
≤ log
E[euX ] · E[evX ] = F (u) + F (v)
which shows convexity of F . The inequality is strict unless there exists some c 6= 0
such that euX = cevX almost surely. This can only be the case if X is con-
stant a.s., in which case F is linear. The same argument applies to R. Property
(iv) follows from the convexity and from the fact that F and R are analytic on
{u ∈ C : Reu ∈ (c1, c2)} (cf. Lukacs [1960, Chapter 7]). �
3. Theoretical Results
We will now use the theory from the last section to calculate bond prices, yields
and other quantities in an interest rate model where the short rate follows a one-
dimensional regular affine process (rt)t≥0 under the martingale measure. Naturally
we will also make the assumption that (rt)t≥0 is conservative, i.e. that pt(x,D) = 1
for all (t, x) ⊆ R>0 ×D. This implies by Duffie et al. [2003, Proposition 9.1] that
c = γ = 0 in Definition 2.3. We will need some additional assumptions which are
summarized in the following condition:
Condition 3.1. The one-dimensional affine process (rt)t≥0 is assumed to be reg-
ular and conservative. In addition, if the process has state space R, such that by
6 MARTIN KELLER-RESSEL AND THOMAS STEINER
Definition 2.3 R(u) = βu, we require that
(3.1) F (u) <∞ for all u ∈
(1/β, 0] if β < 0 ,
(−∞, 0] else .
It will be seen that the condition on F is necessary to guarantee existence of bond
prices for all maturities in the term structure model. By Sato [1999, Theorem 25.17]
we get an equivalent formulation of Condition 3.1, if we replace F (u) < ∞ by∫
|ξ|>1
euξm(dξ) <∞. Next we define a quantity that will generalize the coefficient
of mean reversion from OU-type processes:
Definition 3.2 (quasi-mean-reversion). Given a one-dimensional conservative affine
process (rt)t≥0, define the quasi-mean-reversion λ as the positive solution of
(3.2) R(−1/λ) = 1 .
If there is no positive solution we set λ = 0.
Since R is by Lemma 2.6 a convex function satisfying R(0) = 0, it is easy to see
that (3.2) can have at most one solution and thus λ is well-defined. The name quasi-
mean-reversion is derived from the fact that if (rt)t≥0 is a process of OU-type with
positive mean-reversion, then R(u) = βu and the quasi-mean-reversion λ = −β
is exactly the coefficient of mean reversion of (rt)t≥0. When the process (rt)t≥0
satisfies Condition 3.1, it is seen that F must be defined at least on (−1/λ, 0].
We will encounter several times the condition that λ > 0. The next result gives an
equivalent formulation in terms of (α, β, µ):
Proposition 3.3. The quasi-mean reversion λ is strictly positive if and only if
α > 0,
D\{0}
hR(ξ)µ(dξ) = ∞, or β −
D\{0}
hR(ξ)µ(dξ) < 0.
Proof. First note that by Lemma 2.6 R(u) < ∞ for all u ∈ (−∞, 0]. Using the
estimate (2.7) and a dominated convergence argument it is seen from (2.3) that
= α(3.3)
R(u)− αu2
= β0 := β −
D\{0}
hR(ξ)µ(dξ) ,(3.4)
where β0 can also take the value −∞. Suppose now that α > 0. Then by (3.3) we
get limu→−∞R(u) = ∞. Since R(0) = 0 and R is continuous it follows that there
exists a λ > 0 such that R(−1/λ) = 1. Similarly if α = 0, but β0 < 0, it follows
from (3.4) that limu→−∞R(u) = ∞ and thus again that λ > 0.
Conversely, suppose that α = 0 and β0 ≥ 0. Then
R′(u) = lim
= β0 ≥ 0 .
By the convexity of R it follows that R′(u) ≥ 0 for all u ∈ (−∞, 0). Since R(0) = 0
this implies that R(u) ≤ 0 for all u ∈ (−∞, 0), and consequently that λ = 0. �
3.1. Bond Prices. We consider now the price P (t, t + x) of a zero-coupon bond
with time to maturity x, at time t, given by
P (t, t+ x) = E
∫ t+x
rs ds
)∣∣∣∣Ft
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 7
The affine structure of (rt)t≥0 carries over to the bond prices, and we get the
following result:
Proposition 3.4. Let the short rate be given by a one-dimensional affine process
(rt)t≥0 satisfying Condition 3.1.
Then the bond price P (t, t+ x) exists for all t, x ≥ 0 and is given by
(3.5) P (t, t+ x) = exp (A(x) + rtB(x))
where A and B solve the generalized Riccati equations
∂xA(x) = F (B(x)) A(0) = 0 ,(3.6a)
∂xB(x) = R (B(x)) − 1 B(0) = 0 .(3.6b)
Proof. If D = R>0 the assertion follows directly from Proposition 2.5 by noting
that P (t, t+ x) = Qx 1.
If D = R then, as discussed after Theorem 2.4, (rt)t≥0 is a process of OU-type and
R(u) has the simple structure R(u) = βu. By Sato [1999, (17.2) - (17.3)] we obtain
in this case directly that
(3.7) E
∫ t+x
rs ds
= exp
F (B(s)) ds + rtB(x)
with B(x) = (1 − eβx)/β if β 6= 0 and B(x) = −x when β = 0. As a function of
x ∈ R>0, B is continuously decreasing from 0 to 1/β if β < 0, and from 0 to −∞
if β ≥ 0. It is therefore seen that the integral on the right side of (3.7) is finite for
all x ∈ R>0 if and only if F satisfies (3.1), as required by Condition 3.1. �
Corollary 3.5. Let (rt)t≥0 satisfy Condition 3.1 and have quasi-mean-reversion
λ. Then the function B(x) from Proposition 3.4 is strictly decreasing and satisfies
B(x) = −1/λ .
Proof. The result follows from a qualitative analysis of the autonomous ODE (3.6b).
Let λ > 0. Since R(−1/λ)−1 = 0 the point x∗ := −1/λ is an critical point of (3.6b).
By the convexity of R and the fact that R(0) = 0 it follows that R′(x∗) < 0 such
that x∗ is asymptotically stable, i.e. solutions entering a small enough neighborhood
of x∗ must converge to x∗. Since R(x)− 1 < 0 for x ∈ (x∗, 0] and there is no other
critical point in (x∗, 0], we conclude that B(x) – the solution of (3.6b) starting at
0 – is strictly decreasing and converges to x∗.
If λ = 0 then there is no critical point in (−∞, 0] and R(x)−1 < 0 for x ∈ (−∞, 0].
It follows that B(x) is strictly decreasing and diverges to −∞. �
3.2. The Yield Curve and the Forward Rate Curve. The next results are the
central theoretical results of this article and describe the global shapes of attainable
yield curves in any affine one-factor term structure model.
Definition 3.6. The (zero-coupon) yield Y (rt, x) is given by Y (rt, 0) := rt and
(3.8) Y (rt, x) := −
logP (t, t+ x)
= −A(x)
for all x > 0 .
For rt fixed, we call the function Y (rt, .) the yield curve.
The (instantaneous) forward rate f(rt, x) is given by f(rt, 0) := rt and
(3.9) f(rt, x) := −∂x logP (t, t+ x) = −A′(x)− rtB′(x) for all x > 0 .
For rt fixed, we call the function f(rt, .) the forward rate curve.
8 MARTIN KELLER-RESSEL AND THOMAS STEINER
By l’Hospital’s rule and the generalized Riccati equations (3.6) it is seen that
both the yield and the forward rate curve are continuous at 0.
The first quantity associated to the yield curve that we consider, is the asymptotic
level basymp of the yield curve as x → ∞, also known as long-term yield, consol
yield or simply ‘long end’.
Theorem 3.7. Let the short rate process be given by a one-dimensional affine pro-
cess (rt)t≥0 satisfying Condition 3.1 with quasi-mean-reversion λ.
If λ > 0 then
basymp := lim
Y (rt, x) = lim
f(rt, x) = −F (−1/λ) .
If λ = 0 then
basymp = lim
−F (u) + rt (1−R(u)) .
Proof. From (3.6a) we obtain that
(3.10) lim
= lim
A′(x) = lim
F (B(x)) .
If λ > 0 then by Corollary 3.5
(3.11) lim
B(x) = −1/λ, lim
= 0 and lim
B′(x) = 0
and the assertion follows by combining (3.8) – (3.11).
If λ = 0 then limx→∞B(x) = −∞ and
= lim
B′(x) = lim
R(B(x)) − 1 .
By setting u := B(x) we obtain the desired result. �
From Theorem 3.7 it is clear that for practical purposes only models with λ > 0
will be useful. So far we know that in this case the short end of the yield curve is
given by Y (rt, 0) = rt and the long end by Y (rt,∞) = basymp. We will now examine
what happens between these two endpoints.
Definition 3.8. The yield curve Y (rt, x) is called
• normal if it is a strictly increasing function of x,
• inverse if it is a strictly decreasing function of x,
• humped if it has exactly one local maximum and no minimum on (0,∞).
In addition we call the yield curve flat if it is constant over all x ∈ R>0.
This is our main result on the shapes of yield curves in affine one-factor models:
Theorem 3.9. Let the risk-neutral short rate process be given by a one-dimensional
affine process (rt)t≥0 satisfying Condition 3.1 and with quasi-mean-reversion λ > 0.
In addition suppose that F 6= 0 and that either F or R is non-linear. Then the
following holds:
• The yield curve Y (rt, .) can only be normal, inverse or humped.
• Define
bnorm := −
F ′(−1/λ)
R′(−1/λ) and binv :=
R′(0)
if R′(0) < 0
+∞ if R′(0) ≥ 0 .
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 9
The yield curve is normal if rt ≤ bnorm , humped if bnorm < rt < binv and
inverse if rt ≥ binv .
The above theorem is visualized in Figure 1. For its proof we will use a simple
Lemma. We state the Lemma without proof, since it follows in an elementary way
from the usual definition of a convex function on R.
Lemma 3.10. A strictly convex or a strictly concave function on R intersects an
affine function in at most two points. In the case of two intersection points p1 < p2,
the convex function lies strictly below the affine function on the interval (p1, p2); if
the function is concave it lies strictly above the affine function on (p1, p2).
Proof of Theorem 3.9. Define the function H(x) : R>0 → R by
(3.12) H(x) := Y (rt, x)x = −A(x)− rtB(x) .
We will see that the convexity behavior of H will be crucial for the shape of the
yield curve Y (rt, .). From the generalized Riccati equations (3.6) the first derivative
of H is calculated as
(3.13) ∂xH(x) = −F (B(x)) − rt (R(B(x))− 1)
and the second as
(3.14) ∂xxH(x) = −B′(x) (F ′(B(x)) + rtR′(B(x))) .
Note that F and R are continuously differentiable by Lemma 2.6, and also B by
(3.6b), such that the second derivative of H is well-defined and continuous. Since B
is strictly decreasing by Corollary 3.5, the factor −B′(x) is positive for all x ∈ R>0.
The sign of ∂xxH(x) therefore equals the sign of
(3.15) k(x) := F ′(B(x)) + rtR
′(B(x)) .
From the fact that B is decreasing and F and R are convex it is obvious that k
must be decreasing. We will now show that k has at most a single zero in [0,∞):
(a) D = R>0: We have assumed that either F or R is non-linear. By Lemma 2.6
this implies that either F or R is strictly convex, and thus that either F ′ or R′
is strictly increasing. If rt > 0, then it follows that k is strictly decreasing and
thus has at most a single zero. If rt = 0, an additional argument is needed:
It could happen that F is of the form F = bu such that k(x) = b and k is no
longer strictly decreasing. However, by assumption, F 6= 0 such that in this
case k has no zero in [0,∞).
(b) D = R: In this case, by the admissibility conditions in Definition 2.3, we
have necessarily R(u) = βu. Also, since either F or R is non-linear, F must
be non-linear and thus by Lemma 2.6 strictly convex. It follows that k(x) =
F ′(B(x))+rtβ is strictly decreasing and thus has at most a single zero in [0,∞).
We have shown that k is decreasing and has at most a single zero; to determine
whether it has a zero for some value of rt, we consider the two ‘endpoints’ k(0) and
limx→∞ k(x). First we show that
(3.16) k(0) ≥ 0 if and only if rt ≤ binv :=
R′(0)
if R′(0) < 0
+∞ if R′(0) ≥ 0 .
Since B(0) = 0 by Proposition 3.4 it follows that
k(0) = F ′(0) + rtR
′(0) .
10 MARTIN KELLER-RESSEL AND THOMAS STEINER
We distinguish two cases:
(a) If R′(0) < 0 then the assertion (3.16) follows immediately.
(b) Consider the case that R′(0) ≥ 0: Assume that D = R. Then we have
R(u) = βu and R′(0) = β ≥ 0. This, however, stands in contradiction to
our assumption λ > 0, which implies that β = −λ < 0 (cf. Definition 3.2).
Thus we must have D = R>0 and rt ≥ 0; in this case it follows that k(0) ≥ 0,
for all rt ∈ D, and we set binv = +∞.
Next we consider the right end of k(x) and show that
(3.17) lim
k(x) ≤ 0 if and only if rt ≥ bnorm := −
F ′(−1/λ)
R′(−1/λ) .
Since limx→∞B(x) = −1/λ by Corollary 3.5 we have that
(3.18) lim
k(x) = F ′(−1/λ) + rtR′(−1/λ) .
By assumption λ > 0, and by Definition 3.2 it holds that R(−1/λ) = 1. Also
R(0) = 0, and by the mean value theorem
1 = R(−1/λ)−R(0) = − 1
R′(ξ)
for some ξ ∈ (−1/λ, 0). Since R′ is increasing, it follows that R′(−1/λ) ≤ −λ < 0,
and we can deduce (3.17) directly from (3.18).
We summarize our results on the function k so far: k stays negative on (0,∞) if
rt ≥ binv and positive if rt ≤ bnorm. It has a single zero on (0,∞) if and only if
bnorm < rt < binv. If k has a zero on (0,∞), since k is decreasing, the sign of k will
be positive to the left of the zero and negative to the right of the zero.
Since ∂xxH has the same sign as k, the statements above translate in the obvious
way to the convexity behavior of H . We will now use the convexity behavior of H
to derive our results about the yield curve.
Consider the equation
(3.19) H(x) = cx, x ∈ [0,∞)
for some fixed c ∈ R. Since H(0) = 0 this equation has at least one solution,
x0 = 0. If rt ≥ binv then H(x) is strictly concave on [0,∞), and by Lemma 3.10
the equation (3.19) has at most one additional solution x1. Also, when the solution
exists, H(x) crosses cx from above at x1. Similarly if rt ≤ bnorm then H(x) is
strictly convex, and there exists at most one additional solution x2 to (3.19) on
[0,∞). If the solution exists, then cx is crossed from below at x2. In the last case
bnorm < rt < binv, there exists a x∗ – the zero of k(x) – such that H(x) is strictly
convex on (0, x∗) and strictly concave on (x∗,∞). Now there can exist at most two
additional solutions x1, x2 to (3.19) with x1 < x
∗ < x2, such that cx is crossed from
below at x1 and from above at x2.
Because of definition (3.12), every solution to (3.19), excluding x0 = 0, is also a
solution to
(3.20) Y (rt, x) = c, x ∈ (0,∞)
with rt fixed. Also the properties of crossing from above/below are preserved since
x is positive. This means that in the case rt ≥ binv, equation (3.20) has at most a
single solution, or in other words, that every horizontal line is crossed by the yield
curve at most in a single point. If it is crossed, it is crossed from above. This
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 11
implies that Y (x) is a strictly decreasing function of x, or following Definition 3.8,
that the yield curve is inverse. In the case rt ≤ bnorm we have again that (3.20) has
at most a single solution and that every horizontal line is crossed from below by
the yield curve, if it is crossed. In other words, the yield curve is normal. In the
last case of bnorm < rt < binv, the yield curve crosses every horizontal line at most
twice, in which case it crosses first from below, then from above. Thus in this case
the yield curve is humped. �
Corollary 3.11. Under the conditions of Theorem 3.9 the instantaneous forward
rate curve has the same global behavior as the yield curve, i.e.
Y (rt, .) is inverse ⇐⇒ f(rt, .) is strictly decreasing
Y (rt, .) is humped ⇐⇒
f(rt, .) has exactly one local maximum
and no local minimum
Y (rt, .) is normal ⇐⇒ f(rt, .) is strictly increasing .
In the second case the maximum of the forward rate curve is f(rt, x∗), where x∗
solves
(3.21) rt = −
F ′(B(x))
R′(B(x))
, x ∈ (0,∞) .
Proof. This follows from the fact that ∂xH(x) as given in (3.13) is exactly the
forward rate f(rt, x). The derivative of the forward rate is therefore ∂xxH(x),
which is given in (3.14) as
∂xf(rt, x) = ∂xxH(x) = −B′(x) · k(x) .
The factor −B′(x) 6= 0 is always positive, and the possible sign changes and zeroes
of k(x) are discussed in the proof of Theorem 3.9, leading to the stated equivalences.
Equation (3.21) is simply the condition k(x∗) = 0. �
Corollary 3.12. Under the conditions of Theorem 3.9 it holds that
(3.22) bnorm < basymp < binv
whenever the quantities are finite. In addition it holds that
(3.23) D ∩ (bnorm, binv) 6= ∅ .
Remark 3.13. Note that equation (3.23) implies that there is always some rt ∈ D
such that the yield curve Y (rt, .) is humped.
Proof. By the mean value theorem there exists a ξ ∈ (−1/λ, 0) such that
basymp = −F (−1/λ) = F (0)− F (−1/λ) =
F ′(ξ) .
Since F is convex and thus F ′ increasing, it holds that
(3.24)
F ′(−1/λ)
≤ basymp ≤
F ′(0)
Applying the mean value theorem to R, there exists another ξ ∈ (−1/λ, 0) such
1 = R(−1/λ)−R(0) = − 1
R′(ξ) .
12 MARTIN KELLER-RESSEL AND THOMAS STEINER
Since R′ is increasing we deduce that R′(−1/λ) ≤ −λ < 0. Assuming that also
R′(0) < 0 we get
(3.25) − 1
R′(−1/λ) ≤
≤ − 1
R′(0)
Since either F orR is non-linear, one of the functions is strictly convex by Lemma 2.6.
Consequently either both inequalities in (3.24) or in (3.25) are strict. Putting them
together we get
′(−1/λ)
R′(−1/λ) < basymp < −
F ′(0)
R′(0)
proving (3.22) under the assumption that R′(0) < 0.
If R′(0) ≥ 0 then by definition binv = ∞. Equation (3.24) still holds, but in
(3.25) only the left inequality sign remains valid. Together this still proves that
bnorm < basymp and we have shown (3.22).
To prove (3.23) we distinguish two cases:
(a) D = R. In this case it is sufficient to prove−∞ < binv and bnorm <∞. Consider
first binv. If R
′(0) ≥ 0 then by definition binv = ∞ and nothing is to prove.
If R′(0) < 0 then binv = −F ′(0)/R′(0). By convexity F ′(0) > −∞ and the
assertion follows. Consider now bnorm = −F ′(−1/λ)/R′(−1/λ). From (3.25)
we know that R′(−1/λ) ≤ −λ < 0. By convexity F ′(−1/λ) <∞ and it follows
that bnorm <∞.
(b) D = R>0. In this case it is sufficient to prove 0 ≤ bnorm and to apply (3.22).
As above we have that bnorm = −F ′(−1/λ)/R′(−1/λ) and that R′(−1/λ) ≤
−λ < 0. By Definition 2.3
F ′(−1/λ) = b+
(0,∞)
ξe−ξ/λm(dξ)
with b ≥ 0. It follows that F ′(−1/λ) ≥ 0, proving the assertion. �
The last Corollary of this section shows the interesting fact that the occurrence
of a humped yield curve is a necessary and sufficient sign of randomness in the
short rate model:
Corollary 3.14. Let the risk-neutral short rate process be given by a one-dimensional
affine process (rt)t≥0 satisfying Condition 3.1 with F 6= 0 and quasi-mean-reversion
λ > 0. Then the following statements are equivalent:
(i) There exists a rt ∈ D such that Y (rt, .) is flat.
(ii) There exists no rt ∈ D such that Y (rt, .) is humped.
(iii) The short rate process (rt)t≥0 is deterministic.
(iv) F (u) = bu and R(u) = βu.
Proof. Theorem 3.9, together with Corollary 3.12, shows already that ¬(iv) implies
¬(i) and ¬(ii). Also, from the form of the generator in (2.4), it is seen that (iii) and
(iv) are equivalent. It remains to show that (iv) implies (i) and (ii). Proceeding
as in the proof of Theorem 3.9 we obtain instead of (3.15) simply
k(x) = b+ rtβ .
The yield curve will be humped if and only if k has a single (isolated) zero in [0,∞).
Since k is a constant function, this cannot be the case for any rt ∈ D and we have
shown (ii). By the same arguments as in the proof of Theorem 3.9 the yield curve
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 13
binv = −
F’(0)
R’(0)
basymp = − F(− 1 λ)
bnorm = −
F’(− 1 λ)
R’(− 1 λ)
Time to Maturity
Figure 1. This Figure shows a graphical summary of Theorems
3.7 and 3.9, as well as the definitions of the key quantities bnorm,
basymp and binv. In any affine model satisfying the conditions of
Theorem 3.9, the shapes of yield curves will follow the picture
given here. They will be normal if r0 is below bnorm, humped if r0
is between bnorm and binv and inverse if r0 is above binv. Also all
yield curves will tend asymptotically to the same level basymp.
is flat if and only if k is constant and equal to 0. This is the case if rt = − bβ . It
remains to show that rt ∈ D. Note that β = −λ < 0. In particular β 6= 0, such
that for D = R we are already done. If D = R>0 we have by the admissibility
conditions in Definition 2.3 that b ≥ 0. Thus rt = − bβ ≥ 0 and we have shown
(i). �
3.3. The Limit Distribution of an Affine Process. It is well-known that the
Gaussian Ornstein-Uhlenbeck process, for example, converges in law to a limit
distribution and that this distribution is Gaussian. The goal of this section is to
establish a corresponding result for affine processes. While calculating the marginal
distributions of an affine process involves solving the generalized Riccati equations
(2.5), it will be seen that the limit distribution is much easier obtained and can be
determined directly from the functions F and R.
In the interest rate model considered in the preceding section, the short rate follows
an affine process under the martingale measure, such that the results will allow us
to characterize the risk-neutral asymptotic short rate distribution. Often also the
14 MARTIN KELLER-RESSEL AND THOMAS STEINER
limit distribution under the objective measure is of interest, but the affine prop-
erty is in general not preserved by an equivalent change of measure, such that the
results are not directly applicable. Nevertheless, for the sake of tractability, condi-
tions on the measure change can be imposed, such that the model is affine under
both the objective and the risk-neutral measure. (See Nicolato and Venardos [2003]
for an example from option pricing and Cheridito et al. [2005] for more general re-
sults). In such a setting the results can also be applied under the objective measure.
Before we state the result, we want to recall that a real-valued random variable
L is called self-decomposable if for every c ∈ (0, 1) there exists a random variable
Lc, independent of L, such that
L = cL+ Lc for all c ∈ (0, 1) .
Since self-decomposability is a distributional property, we will identify L and its
law, and refer to both as self-decomposable.
For OU-type processes, limit distributions have been studied for some time;
the first results can be found in Jurek and Vervaat [1983] and Sato and Yamazato
[1984]. The next theorem summarizes these results, and can be found in similar
form in Sato [1999, Theorem 17.5]:
Theorem 3.15. Let (rt)t≥0 be a OU-type process on R. If
β < 0 and
|ξ|>1
log |ξ|m(dξ) <∞
then (rt)t≥0 converges in law to a limit distribution L which is independent of r0
and has the following properties:
(i) L is self-decomposable.
(ii) The cumulant generating function κ(u) = log
eux dL(x) satisfies
(3.26) κ(iu) =
F (is)
ds for all u ∈ R .
Conversely, if L is a self-decomposable distribution on R and β < 0, there exists
a unique triplet (a, b,m) satisfying the admissibility conditions of Definition 2.3,
such that L is the limit distribution of the affine process (of OU-type) given by the
parameters (a, b,m, β).
As discussed in Section 2, every regular affine process with state space R is of
OU-type, such that the above theorem applies. We now state our corresponding
result for affine processes on R>0:
Theorem 3.16. Let (rt)t≥0 be a one-dimensional, regular, conservative affine pro-
cess with state space R>0. If
R′(0) < 0 and
log ξ m(dξ) <∞
then (rt)t≥0 converges in law to a limit distribution L which is independent of r0,
and whose cumulant generating function κ is given by
(3.27) κ(u) =
F (s)
ds for all u ∈ (−∞, 0] .
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 15
Proof. By Theorem 2.4 the transition kernel pt(x, .) of the process (rt)t≥0 has the
characteristic function
p̂t(x, u) = exp (φ(t, u) + xψ(t, u))
where φ and ψ satisfy the generalized Riccati equations (2.5) for all u ∈ U , and
thus in particular for all u ∈ (−∞, 0]. Since R(0) = 0, 0 is a critical point of the
autonomous ODE (2.5b), and by the assumption R′(0) < 0 it is asymptotically
stable. By the convexity of R, R′(0) < 0 also implies that R(u) > 0 for all u ∈
(−∞, 0), such that ψ(t, u) is a strictly increasing function in t for all u ∈ (−∞, 0).
Since 0 is the only critical point of (2.5b) on (−∞, 0] it also follows that
ψ(t, u) = 0 for all u ∈ (−∞, 0] .
Consequently,
(3.28) lim
log p̂t(x, u) = lim
φ(t, u) =
F (ψ(r, u)) dr =
F (s)
where the last two equalities follow from (2.5) and the transformation s = ψ(r, u).
We will now show that the last integral in (3.28) converges absolutely for all u ∈
(−∞, 0]: Since R(u) ≥ 0 and F (u) ≤ 0 for all u ∈ (−∞, 0] we obtain
F (s)
∣∣∣∣ ds = −
F (s)
ds ≤ − 1
R′(0)
F (s)
ds, u ∈ (−∞, 0] ,
where the inequality follows from the fact that the convex function R is supported by
its tangent at 0. From the definition of F (u) in (2.2) it is clear that the convergence
of the last integral will depend only on the jump part of F , i.e. the integral converges
if and only if
(3.29)
(0,∞)
esξ − 1
m(dξ) ds <∞, for all u ∈ (−∞, 0].
Define M(u, ξ) =
esξ−1
ds. For a fixed u ∈ (−∞, 0], it is easily verified that
M(u, ξ) = O(ξ) as ξ → 0, and that M(u, ξ) = O(log ξ) as ξ → ∞. Since the Lévy
measurem(dξ) integrates (ξ∧1) by Definition 2.3, and log ξ ·1{ξ>1} by assumption,
it must also integrateM(u, ξ). Applying Fubini’s theorem, (3.29) follows, such that
κ(u) :=
F (s)
ds converges for all u ∈ (−∞, 0]. In particular limu↑0 κ(u) = 0, such
that the limit in (3.28) is a function that is left-continuous at 0. By standard results
on Laplace transforms of probability measures (cf. Steutel and van Harn [2004,
Theorem A.3.1]), the pointwise convergence of cumulant generating functions to a
function that is left-continuous at 0 implies convergence in distribution of (rt)t≥0
to a limit distribution L with cumulant generating function given by (3.28). �
Since the marginal distributions of an affine process are infinitely divisible, also
the limit distribution L must be infinitely divisible, if it exists. In Theorem 3.15
a stronger result is given for an affine process on R: In this case L is also self-
decomposable. An obvious question is, if this result can be extended to the state
space R>0. We will see that the answer is negative. In Section 4.3 an example of an
affine process with state space R>0 is given, which converges to an infinitely divisible
limit distribution that is not self-decomposable. This result is interesting, since it
leaves open the possibility of some unexpected properties of the limit distribution of
16 MARTIN KELLER-RESSEL AND THOMAS STEINER
an affine process. For example a self-decomposable distribution is always unimodal,
whereas an infinitely divisible distribution might be not.
4. Applications
4.1. The Vasiček model. We apply the results of the last section to the classical
Vasiček model
(4.1) drt = −λ(rt − θ) dt+ σ dWt, r0 ∈ R
where (Wt)t≥0 is a standard Brownian motion under the risk-neutral measure and
λ, θ, σ > 0. The Vasiček model is arguably the simplest affine model, and no
surprises are to be expected here. In fact all results that we state here can already
be found in the original paper of Vasiček [1977]. We advise the reader to view this
paragraph as a warm-up for the following examples.
Clearly (rt)t≥0 is a conservative affine process with
F (u) = λθu +
u2 ,(4.2)
R(u) = −λu .(4.3)
From the quadratic term in F and Definition 2.3, it is seen that (rt)t≥0 has state
space R. This property is often criticized, since it allows the short rate to become
negative.
From Theorem 3.9 we calculate
binv = θ and bnorm = θ −
such that the yield curve in the Vasiček model is normal if rt ≤ θ − σ2/λ2, inverse
if rt ≥ θ and humped in the remaining cases.
The long term yield is calculated from (3.7) as
basymp = −F (−1/λ) = θ −
in this case exactly the arithmetic mean of binv and bnorm.
Theorem 3.15 applies and the cumulant generating function κ of the risk-neutral
limit distribution L satisfies
κ(iu) = − 1
F (is)
iθ − σ
ds = uiθ − u
for u ∈ R. Hence, L is Gaussian with mean θ and variance σ2
4.2. The Cox-Ingersoll-Ross model. The Cox-Ingersoll-Ross (CIR)-model was
introduced by Cox et al. [1985]. In this model the short rate process (rt)t≥0 is given
by the SDE
(4.4) drt = −a(rt − θ)dt+ σ
rt dWt, r0 ∈ R>0
where (Wt)t≥0 is a standard Brownian Motion under the risk-neutral measure and
a, θ, σ > 0. The process (rt)t≥0 is a conservative affine process with
F (u) = aθu ,(4.5)
R(u) =
u2 − au .(4.6)
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 17
From Definition 2.3 it is seen that (rt)t≥0 has state space R>0. The fact that
interest rates stay non-negative in the CIR-model is often cited as an advantage
of the model over the Vasiček model. Calculating the quasi-mean-reversion (see
Definition 3.2), we find that
a2 + 2σ2 + a
From Theorem 3.7 we find that the long-term yield is given by
basymp = −F (−1/λ) =
a2 + 2σ2 + a
The boundary between humped and inverse behavior binv is calculated from Theo-
rem 3.9 as
binv = −
F ′(0)
R′(0)
Both quantities basymp and binv can also be found in [Cox et al., 1985, Eq. (26) and
following paragraph]. Before we consider bnorm, we quote (with notation adapted
to (4.4)) from page 394 of [Cox et al., 1985] where the shape of the yield curve is
discussed:
‘When the spot rate is below the long-term yield [= basymp], the
term structure is uniformly rising. With an interest rate in excess
of θ [= binv], the term structure is falling. For intermediate values
of the interest rate, the yield curve is humped.’
In our terminology, they claim that the yield curve is normal for rt ≤ basymp,
humped for basymp < rt < binv and inverse for rt ≥ binv. This stands in clear
contradiction to Theorem 3.9 and Corollary 3.12 where we have obtained that yield
curves are normal if and only if rt ≤ bnorm and that bnorm < basymp, or – in plain
words – that there are yield curves starting strictly below the long-term yield that
are still humped.
The claims of Cox et al. [1985] are repeated in [Rebonato, 1998, p. 244f], where even
several plots of ‘yield surfaces’ (the yield as a function of rt and x) are presented as
evidence. However Rebonato fails to indicate the level of basymp in the plots, such
that the conclusion remains ambiguous.
To clarify the scope of humped yield curves in the CIR-model we calculate bnorm
from Theorem 3.9:
bnorm = −
F ′(−1/λ)
R′(−1/λ) =
a2 + 2σ2
The relation bnorm < basymp < binv is immediately confirmed by noting that basymp
is the harmonic mean of bnorm and binv. For a graphical illustration we refer to
the second yield curve from below in Figure 1. The plot actually shows CIR yield
curves with parameters
a = 0.5, σ = 0.5, θ = 6%
plotted over a time scale of 25 years. The second curve from below starts at
r0 = 4.2%, i.e. below the long-term yield, but is visibly humped.
18 MARTIN KELLER-RESSEL AND THOMAS STEINER
To calculate the limit distribution of (rt)t≥0, we apply Theorem 3.16: The cu-
mulant generating function κ(u) of the limit distribution is given by
κ(u) =
F (s)
1− sσ2/2a ds = −
This is the cumulant generating function of a gamma distribution with shape pa-
rameter 2aθ/σ2 and scale parameter σ2/2a. Again this result can already be found
in Cox et al. [1985, p. 392].
4.3. An extension of the CIR model. To illustrate the power of the affine
setting, we consider now an extension of the CIR model that is obtained by adding
jumps to (4.4). We define the risk-neutral short rate process by
(4.7) drt = −a(rt − θ)dt+ σ
rt dWt + dJt, r0 ≥ 0
where (Jt)t≥0 is a compound Poisson process with intensity c > 0 and expo-
nentially distributed jumps of mean ν > 0. This model has been introduced by
Duffie and Gârleanu [2001] as a model for default intensity and is used by Filipović
[2001] as a short rate model. It can also be found in Brigo and Mercurio [2006]
under the name JCIR model. It is easily calculated that
F (u) = aθu+
ν − u, u ∈ (−∞, ν) ,(4.8)
R(u) =
u2 − au .(4.9)
Solving the generalized Riccati equations (3.6) for A(x) and B(x) becomes quite
tedious, but the quantities binv, basymp, bnorm can be calculated from Theorem 3.7
and Theorem 3.9 in a few lines: The quasi-mean reversion λ stays the same as in
the CIR model, since R does not change. From
F ′(u) = aθ +
(ν − u)2
we derive immediately
binv = θ +
basymp =
ν(a+ ν) + 2
bnorm =
γ(σ2ν + γ − a)2 ,
where γ =
a2 + 2σ2. Note that by setting the jump intensity c to zero, the ex-
pressions of the (original) CIR model are recovered.
Next we calculate the limit distribution of the model. Using the abbreviations
ρ := σ2/2 and ∆ := a− νρ we obtain
κ(u) =
F (s)
1− sρ/a ds+ c
(s− ν)(ρs− a) =
if ∆ 6= 0
−θν log
if ∆ = 0
YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 19
as the cumulant generating function of the limit distribution L under the martin-
gale measure.
We now take a closer look at the distribution L, since this will answer the
question raised at the end of Section 3.3: For certain parameters, L is an example
for a limit distribution of an affine process that is infinitely divisible, but not self-
decomposable. We consider the case ∆ = 0 and define
(4.10) l(x) :=
νe−νx, x ∈ R>0 .
By Frullani’s integral formula
(4.11) κ(u) =
(eux − 1) l(x)
for all u ∈ (−∞, ν). Since l is non-negative on R>0, l(x)/x is the density of a
Lévy measure and (4.11) is seen to be the Lévy-Khintchine representation for the
cumulant generating function of the infinitely divisible distribution L. In addition,
L is self-decomposable if and only if l is non-negative and non-increasing on R>0
(cf. Sato [1999, Corollary 15.11]).
In the case of l(x) given by (4.10), it is easily calculated that l(x) has a single
maximum at x∗ = 1
. Thus, if c ≤ aθν, then x∗ ≤ 0, such that l is non-
increasing on R>0 and L is self-decomposable. If c > aθν then l is increasing
in the interval [0, x∗) and the limit distribution L is infinitely divisible, but not
self-decomposable.
4.4. The gamma model. Instead of analyzing the properties of a known model,
we will now follow a different route and construct a model that satisfies some given
properties. We want to construct an affine process on R>0 that has the same limit
distribution as the CIR model (i.e. a gamma distribution), but is a process of OU-
type. The second property is equivalent to R(u) = βu. Considering Theorem 3.16,
we know that if we want to obtain a limit distribution, we need β < 0. To keep
with the notation of the Vasiček model, we will write R(u) = −λu where λ > 0.
Now by (3.27) the cumulant generating function of the limit distribution is given
(4.12) κ(u) =
F (s)
ds for all u ∈ (−∞, 0] .
Let the limit distribution be a gamma distribution with shape parameter k > 0
and scale parameter θ > 0. Then κ(u) = −k log(1 − θu) and by (4.12)
F (u) =
1− θu .
Setting c = λk and ν = 1/θ it is seen that F (u) is equal to the last term in
(4.8). This means that the driving Lévy process of (rt)t≥0 is of the same kind
as the process (Jt)t≥0 in (4.7), i.e. (rt)t≥0 is a pure jump OU-type process with
exponentially distributed jump heights of mean 1/θ and with jump intensity λk.
We interpret the affine process we have constructed as a risk-neutral short rate
process. It is clear that the bond prices are of the exponentially-affine form (3.5).
From the generalized Riccati equation (3.6b) we obtain
B(x) =
e−λx − 1
20 MARTIN KELLER-RESSEL AND THOMAS STEINER
From equation (3.6a) we calculate
A(x) =
F (B(s)) ds =
θ + λ
(log(1− θB(x)) − θx) ,
such that the bond prices are given by
P (t, t+ x) = exp
−x λθk
θ + λ
+ rtB(x)
(1− θB(x))
θ+λ .
The global shape of the yield curve is described by the quantities
binv = kθ, basymp =
1/θ+ 1/λ
, bnorm =
(1/θ + 1/λ)2
and it is seen that for the gamma-OU-process basymp is the geometric average of
binv and bnorm.
5. Conclusions
In this article we have given, under very general conditions, a characterization of
the yield curve shapes that are attainable in term structure models where the risk-
neutral short rate is given by a time-homogenous, one-dimensional affine process.
Even though the parameter space for this class of models is infinite-dimensional,
the scope of attainable yield curves is very narrow, with only three possible global
shapes. In addition we have given conditions under which an affine process con-
verges to a limit distribution, and we have characterized the limit distribution in
terms of its cumulant generating function, extending some known results on OU-
type processes.
The most obvious question for future research is the extension of these results to
multi-factor models. It is evident from numerical results that in two-factor models
yield curves with e.g. a dip, or also with a dip and a hump, can be obtained. It
would be interesting to see if more complex shapes can also be produced, or if there
are similar limitations as in the single-factor case. Also, in the one-factor case the
dependence of the yield curve shape on the current short rate is basically described
by the intervals D ∩ (−∞, bnorm], (bnorm, binv) and [binv,∞). In the two-factor case
the partitioning of the state-space might be more complex, and we expect to see
more interesting transitions between yield curve types. Another aspect is, that
since affine processes as a general framework become better understood, extensions
of classical models e.g. by adding jumps, like in the JCIR model described in
Section 4.3, become more feasible and attractive for applications.
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Vienna University of Technology, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria
E-mail address: mkeller@fam.tuwien.ac.at
Vienna University of Technology, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria
E-mail address: thomas@fam.tuwien.ac.at
	1. Introduction
	2. Preliminaries
	3. Theoretical Results
	3.1. Bond Prices
	3.2. The Yield Curve and the Forward Rate Curve
	3.3. The Limit Distribution of an Affine Process
	4. Applications
	4.1. The Vasicek model
	4.2. The Cox-Ingersoll-Ross model
	4.3. An extension of the CIR model
	4.4. The gamma model
	5. Conclusions
	References
