Correction to: Yield curve shapes and the asymptotic short rate distribution in affine one-factor models

Correction to: Finance Stoch. (2008) 12: 149–172 https://doi.org/10.1007/s00780-007-0059-z

I should like to thank Ralf Korn for alerting me to an error in the original paper [2]. The error concerns the threshold at which the yield curve in an affine short rate model changes from normal (strictly increasing) to humped (endowed with a single maximum). In particular, it is not true that this threshold is the same for the forward curve and for the yield curve, as claimed in [2]. Below, the correct mathematical expression for the threshold is given, supplemented with a self-contained and corrected proof.

1 Setting

In [2], affine short rate models for bond pricing were considered, i.e., models where the risk-neutral short rate process $r = (r_{t})_{t \geq 0}$ is given by an affine process in the sense of [1]. The process $r$ takes values in a state space $D$, which is either ${[0,\infty )}$ or ℝ. In this setting, the price at time $t$ of a zero-coupon bond with time to maturity $x$, denoted by $P(t,t+x)$, is of the form

$$P(t,t+x) = \exp \big(A(x) + r_{t} B(x)\big), $$

where $A$ and $B$ satisfy the generalized Riccati differential equations

$$ \begin{aligned} \partial _{x} A(x) &= F\big(B(x)\big), & \qquad A(0) &= 0,\\ \partial _{x} B(x) &= R\big(B(x)\big) -1, & \qquad B(0) &= 0. \end{aligned} $$

(1.1)

The functions $F$ and $R$ are of Lévy–Khintchine form and their parametrization is in one-to-one correspondence with the infinitesimal generator of $r$; cf. [2, Sect. 2]. Derived from the bond price are the yield curve

$$Y(x,r_{t}) := -\frac{\log P(t,t+x)}{x} = -\frac{A(x)}{x} - r_{t} \frac{B(x)}{x} $$

and the forward curve

$$f(x,r_{t}) := -\partial _{x} \log P(t,t+x) = -A'(x) - r_{t} B'(x). $$

The first objective of [2] was to derive the long-term yield and long-term forward rate. It was shown that the equation $R(c) = 1$ has at most a single negative solution $c$, and that under mild conditions,

$$b_{\mathrm{asymp}}:= \lim _{x \to \infty } Y(x,r_{t}) = \lim _{x \to \infty } f(x,r_{t}) = -F(c) $$

if such a solution exists; cf. [2, Theorem 3.7]. We remark that $\lambda := -\frac{1}{c} > 0$ was called quasi-mean-reversion of $r$ in [2], with the convention that $\lambda = 0$ if no negative solution $c$ exists. The second objective of [2] was to characterize all possible shapes of the yield and the forward curve. Recall that in common terminology, the yield or the forward curve 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 local minimum in $(0,\infty )$.

Finally, we recall the technical condition [2, Condition 3.1] in slightly rephrased form. The condition is necessary to guarantee finite bond prices when negative values of the short rate are allowed.

Condition 1.1

We assume that $r$ is regular and conservative. If $r$ has state space $D = \mathbb{R}$, which necessarily implies that $R$ is of the linear form $R(x) = \beta x$ (cf. [1]), we require that

$$F(x) < \infty \qquad \mbox{for all } x \in \textstyle\begin{cases} (1/\beta ,0], &\quad \mbox{if}\ \beta < 0, \\ (-\infty ,0], &\quad \mbox{else}. \end{cases} $$

2 Corrections to results

Theorem 3.1 in [2] should be replaced by the following corrected version.

Theorem 2.1

Let the risk-neutral short rate be given by a one-dimensional affine process $(r_{t})_{t \geq 0}$ satisfying Condition 1.1 and with quasi-mean-reversion $-1/c = \lambda > 0$. In addition, suppose that $F \neq 0$ and that at least one of $F$ and $R$ is nonlinear. Then the following hold:

• (1) The yield curve $Y(\cdot ,r_{t})$ can only be normal, inverse or humped.

• (2) Define

  $$\begin{aligned} b_{\mathrm{y}\text{-}\mathrm{norm}} &:= \frac{1}{c}\int _{c}^{0} \frac{F(u) - F(c)}{R(u) - 1} du, \qquad \\ b_{\mathrm{inv}} &:= \textstyle\begin{cases} -\frac{F'(0)}{R'(0)}, &\quad \textit{if}\ R'(0) < 0, \\ +\infty , &\quad \textit{if}\ R'(0) \ge 0. \end{cases}\displaystyle \end{aligned}$$

  The yield curve is normal if $r_{t} \le b_{\mathrm{y}\text{-}\mathrm{norm}}$, humped if $b_{\mathrm{y}\text{-}\mathrm{norm}}< r_{t} < b_{\mathrm{inv}}$, and inverse if $r_{t} \ge b_{\mathrm{inv}}$.

Remark 2.2

The correction only concerns the expression for $b_{\mathrm{y} \text{-norm}}$, which was called $b_{\mathrm{norm}}$ in [2] and erroneously given as $b_{\mathrm{norm}}= -F'(c) / R'(c)$. All other parts of the theorem are the same as in [2, Theorem 3.1].

Corollary 3.11 in [2] should be replaced by the following result.

Theorem 2.3

Define $b_{\mathrm{inv}}$ as in Theorem 2.1 and set

$$b_{\mathrm{fw}\text{-}\mathrm{norm}}:= -\frac{F'(c)}{R'(c)}. $$

Under the conditions of Theorem 2.1, the following hold:

• (1) The forward curve $f(\cdot ,r_{t})$ can only be normal, inverse or humped.

• (2) The forward curve is normal if $r_{t} \le b_{\mathrm{fw}\text{-}\mathrm{norm}}$, humped if $b_{\mathrm{fw}\text{-}\mathrm{norm}} < r_{t} < b_{\mathrm{inv}}$, and inverse if $r_{t} \ge b_{\mathrm{inv}}$.

Remark 2.4

We have intentionally renamed the result from corollary to theorem, since the correction changes the logical structure of the proof. Note that the above result is equivalent to [2, Corollary 3.11] up to the notational change from $b_{\mathrm{norm}}$ to $b_{\mathrm{fw}\text{-}\mathrm{norm}}$. Note that $b_{\mathrm{y}\text{-}\mathrm{norm}}\neq b_{\mathrm{fw}\text{-}\mathrm{norm}}$ in general, while in [2] it was erroneously claimed that $b_{\mathrm{y}\text{-}\mathrm{norm}}= b_{\mathrm{fw} \text{-}\mathrm{norm}}$.

Corollary 3.12 in [2] should be replaced by the following result.

Corollary 2.5

Under the conditions of Theorem 2.1, it holds that

$$ b_{\mathrm{fw}\text{-}\mathrm{norm}}< b_{\mathrm{y}\text{-}\mathrm{norm}}< b_{ \mathrm{asymp}}< b_{\mathrm{inv}}. $$

(2.1)

In addition, the state space $D$ of the short rate process satisfies

$$D \cap (b_{\mathrm{y}\text{-}\mathrm{norm}}, b_{\mathrm{inv}}) \neq \emptyset . $$

The error also affects [2, Fig. 1], where the expression for $b_{\mathrm{norm}}$ should be replaced by the correct value of $b_{\mathrm{y}\text{-norm}}$. It also affects the application section [2, Sect. 4], where the values of $b_{\mathrm{norm}}$ and $b_{\mathrm{inv}}$ are calculated in different models. The corrections to [2, Sect. 4] are as follows.

In the Vasiček model, the short rate is given by

$$dr_{t} = -\lambda (r_{t} - \theta )\,dt + \sigma \,dW_{t}, \quad r _{0} \in \mathbb{R}, $$

with $\lambda , \theta , \sigma > 0$. This leads to the parametrization

$$\begin{aligned} F(u) &= \lambda \theta u + \frac{\sigma ^{2}}{2}u^{2}, \\ R(u) &= -\lambda u. \end{aligned}$$

By direct calculation, we obtain

$$\begin{aligned} b_{\mathrm{y}\text{-}\mathrm{norm}} &= \theta - \frac{3 \sigma ^{2}}{4 \lambda ^{2}}, \\ b_{\mathrm{fw}\text{-}\mathrm{norm}} &= \theta - \frac{\sigma ^{2}}{\lambda ^{2}}. \end{aligned}$$

Note that the value of $b_{\mathrm{y}\text{-}\mathrm{norm}}$ is now consistent with the results of [3, p. 186].

In the Cox–Ingersoll–Ross model, the short rate is given by

$$r_{t} = - a(r_{t} - \theta )\,dt + \sigma \sqrt{r_{t}} \,dW_{t}, \quad r_{0} \in {[0,\infty )}, $$

with $a, \theta , \sigma > 0$. This leads to the parametrization

$$\begin{aligned} F(u) &= a \theta u, \\ R(u) &= - \frac{\sigma ^{2}}{2}u^{2} - au. \end{aligned}$$

By direct calculation, we obtain

$$\begin{aligned} b_{\mathrm{y}\text{-}\mathrm{norm}} &= \frac{2 a \theta }{\gamma - a} \log \frac{2 \gamma }{a + \gamma }, \\ b_{\mathrm{fw}\text{-}\mathrm{norm}} &= \frac{a \theta }{\gamma }, \end{aligned}$$

where $\gamma := \sqrt{2\sigma ^{2} + a^{2}}$.

In the gamma model, the short rate is given by an Ornstein–Uhlenbeck-type process, driven by a compound Poisson process with intensity $\lambda k$ and exponentially distributed jump heights of mean $1/\theta$; see [2, Sect. 4.4] for details. In this model, we have

$$\begin{aligned} F(u) &= \frac{\lambda \theta k u}{1 - \theta u}, \qquad R(u) = - \lambda u, \end{aligned}$$

and by direct calculation, we obtain

$$\begin{aligned} b_{\mathrm{y}\text{-norm}} &= \frac{k \lambda }{1 + \theta /\lambda } \log (1 + \theta /\lambda ), \\ b_{\mathrm{fw}\text{-}\mathrm{norm}} &= \frac{k \theta }{(1 + \theta /\lambda )^{2}}. \end{aligned}$$

Since the resulting expressions are quite involved, we omit the calculations for the extended CIR model [2, Eq. (4.7)].

3 Corrected proofs

To prepare for the corrected proofs, we collect the following properties from [2, Sects. 2 and 3.1], which hold for the functions $F$, $R$, $B$ and for the state space $D$ under the assumptions of Theorem 2.1:

• (1) $F$ is either strictly convex or linear; the same holds for $R$. Both functions are continuously differentiable on the interior of their effective domains.

• (2) The function $B$ is strictly decreasing with limit $\lim _{x \to \infty } B(x) = c$.

• (3) $F(0) = R(0) = 0$ and $R'(c) < 0$. In addition, $F'(0) >0$ if $D = {[0,\infty )}$.

• (4) Either

  • a) $D = {[0,\infty )}$, or

  • b) $D = \mathbb{R}$ and $R(u) = u/c$ with $c < 0$.

Note that Theorem 2.1 assumes that at least one of $F$ and $R$ is nonlinear. Together with (P1), this implies

• (1) At least one of $F$ and $R$ is strictly convex.

In addition, we introduce the following terminology. Let $f: (0, \infty ) \to \mathbb{R}$ be a continuous function. The zero set of $f$ is $Z := \left \{ x \in (0,\infty ): f(x) = 0\right \}$. The sign sequence of $Z$ is the sequence of signs $\left \{ +,-\right \}$ that $f$ takes on the complement of $Z$, ordered by the natural order on ℝ. For example, the function $x^{2} - 1$ on $(0,\infty )$ has the finite sign sequence $(-+)$; the function $\sin x$ has the infinite sign sequence $(+-+-\cdots )$. An obvious, but important property is the following: Let $g: (0,\infty ) \to (0,\infty )$ be a positive continuous function. Then $fg$ has the same zero set and the same sign sequence as $f$.

Proof of Theorem 2.3

From the Riccati equations (1.1), we can write the derivative of the forward curve as

$$ \partial _{x} f(x,r_{t}) = - B'(x) \underbrace{\Big(F'\big(B(x)\big) + r_{t} R'\big(B(x)\big)\Big)}_{=:k(x)}. $$

(3.1)

Note that by (P2), the factor $-B'(x)$ is strictly positive, and hence $\partial _{x} f$ has the same sign sequence as $k$. We distinguish cases (a) and (b) as in (P4).

(a) Assume that $r_{t} \in D = {[0,\infty )}$. By (P2), $B(x)$ is strictly decreasing, and by (P1′), either $F'$ or $R'$ is strictly increasing. Thus if $r_{t} > 0$, it follows that $k(x)$ is a strictly decreasing function. If $r_{t} = 0$, then $k$ is either strictly decreasing (if $F'$ is strictly convex) or $k$ is constant (if $F$ is linear). By (P1), these are the only possibilities. In addition, the case $F = 0$ is ruled out by the assumptions.

(b) Assume that $r_{t} \in D = \mathbb{R}$. In this case, $R(u) = u/c$, and hence $R'(u) = 1/c$ is constant and $F'$ is strictly increasing, by (P1′). We conclude that $k$ is strictly decreasing.

In any case, $k$ is either strictly decreasing or constant and non-zero. Thus the sign sequence of $k$ can be completely characterized by its initial value $k(0)$ and its asymptotic limit as $x$ tends to infinity. Let us first show that

$$ k(0) \le 0 \quad \Longleftrightarrow \quad r_{t} \ge b_{\mathrm{inv}}= \textstyle\begin{cases} -\frac{F'(0)}{R'(0)}, &\quad \text{if $R'(0) < 0$}, \\ +\infty , &\quad \text{if $R'(0) \ge 0$}. \end{cases} $$

(3.2)

Because we have $k(0) = F'(0) + r_{t} R'(0)$, the assertion follows immediately if $R'(0) < 0$. Consider the complementary case $R'(0) \ge 0$. This rules out case (b) in (P4), and hence we may assume that $D = {[0,\infty )}$. Since $F'(0) > 0$ by (P3), (3.2) follows. Next we show that

$$ \lim _{x \to \infty }k(x) \ge 0 \quad \Longleftrightarrow \quad r_{t} \le b_{\mathrm{fw}\text{-}\mathrm{norm}}= -\frac{F'(c)}{R'(c)} . $$

(3.3)

This follows immediately from $\lim _{x \to \infty }k(x) = F'(c) + r _{t} R'(c)$ and $R'(c) < 0$, by (P3). Combining (3.2) with (3.3) and using that $k$ is either strictly decreasing or constant and non-zero, we obtain

$$ \begin{aligned} r_{t} \ge b_{\mathrm{inv}}\quad & \Longleftrightarrow \quad k \text{ has sign sequence } (-), \\ r_{t} \le b_{\mathrm{fw}\text{-}\mathrm{norm}}\quad & \Longleftrightarrow \quad k \text{ has sign sequence } (+), \\ r_{t} \in (b_{\mathrm{fw}\text{-}\mathrm{norm}}, b_{\mathrm{inv}}) \quad & \Longleftrightarrow \quad k \text{ has sign sequence } (+-). \end{aligned} $$

(3.4)

Since $\partial _{x} f$ has the same sign sequence as $k$, these statements can be directly translated into monotonicity properties of $f$. In the first case, the forward curve $f$ is strictly decreasing, i.e., inverse; in the second case, it is strictly increasing, i.e., normal. In the third case, it is strictly increasing up to the unique zero of $k$ and then strictly decreasing, i.e., humped. No other cases are possible. □

Proof of Theorem 2.1

From the Riccati equations (1.1), we can write the derivative of the yield curve as

$$\partial _{x} Y(x,r_{t}) = \frac{1}{x^{2}} \big(A(x) + r_{t} B(x) \big) - \frac{1}{x} \bigg(F\big(B(x)\big) + r_{t} \Big(R\big(B(x) \big) - 1\Big)\bigg). $$

Multiplying by the positive function $x^{2}$, we see that $\partial _{x} Y(x,r_{t})$ has the same zero set and the same sign sequence as

$$M(x) := \Big(A(x) - x F\big(B(x)\big)\Big) + r_{t} \bigg(B(x) - x \Big(R\big(B(x)\big) - 1\Big)\bigg). $$

The derivative of $M$ is given by

$$M'(x) := -x B'(x) \Big(F'\big(B(x)\big) + r_{t} R'\big(B(x)\big) \Big) = - xB'(x) k(x), $$

with $k$ as in (3.1). Note that by (P2), the factor $-x B'(x)$ is strictly positive, and hence $M'$ has the same sign sequence as $k$, which was already analyzed in (3.4). Since $M(0) = 0$, we can conclude that

$$ \begin{aligned} r_{t} \ge b_{\mathrm{inv}}\quad & \Longrightarrow \quad M \text{ has sign sequence } (-), \\ r_{t} \le b_{\mathrm{fw}\text{-}\mathrm{norm}}\quad & \Longrightarrow \quad M \text{ has sign sequence } (+), \\ r_{t} \in (b_{\mathrm{fw}\text{-}\mathrm{norm}}, b_{\mathrm{inv}}) \quad & \Longrightarrow \quad M \text{ has sign sequence } (+-) \text{ or } (+). \end{aligned} $$

(3.5)

Essentially, the mistake in [2] was to ignore the possible sign sequence $(+)$ in the third case. Not repeating the same mistake, we take a closer look at the third case and note that the sign sequence of $M$ is $(+-)$ if and only if

$$ \lim _{x \to \infty } M(x) < 0. $$

Decomposing $M(x) = L_{1}(x) + r_{t} L_{2}(x)$, it remains to study the asymptotic properties of $L_{1}$ and $L_{2}$. We have

$$\begin{aligned} L_{1}(x) &= A(x) - x F\big(B(x)\big) = \int _{0}^{x} \Big(F\big(B(s) \big) - F\big(B(x)\big)\Big)ds \\ &= \int _{0}^{B(x)} \frac{F(u) - F(B(x))}{R(u) - 1}du \quad \xrightarrow{x \to \infty } \quad \int _{0}^{c} \frac{F(u) - F(c)}{R(u) - 1} du. \end{aligned}$$

In addition,

$$\begin{aligned} L_{2}(x) &= B(x) - x \Big(R\big(B(x)\big) - 1\Big) = \int _{0}^{x} \Big(R\big(B(s)\big) - R\big(B(x)\big)\Big)ds \\ &= \int _{0}^{B(x)} \frac{R(u) - R(B(x))}{R(u) - 1}du \quad \xrightarrow{x \to \infty } \quad \int _{0}^{c} \frac{R(u) - 1}{R(u) - 1} du = c. \end{aligned}$$

Since $c < 0$, we conclude that

$$ \lim _{x \to \infty }M(x) < 0 \quad \Longleftrightarrow \quad r_{t} > b _{\mathrm{y}\text{-norm}}= \frac{1}{c}\int _{c}^{0} \frac{F(u) - F(c)}{R(u) - 1} du. $$

By convexity of $F$ and $R$ and using that $c < 0$, we observe that

$$ b_{\mathrm{y}\text{-norm}}= \frac{1}{c}\int _{c}^{0} \frac{F(u) - F(c)}{R(u) - 1} du \ge \frac{1}{c}\int _{c}^{0} \frac{F'(c)}{R'(c)} du = - \frac{F'(c)}{R'(c)} = b_{\mathrm{fw}\text{-}\mathrm{norm}}. $$

Together with (3.5), this completes the proof. □

Proof of Corollary 2.5

Recall that $R(c) = 1$ and $c < 0$. By convexity of $F$ and $R$, we have

$$ \begin{aligned} F'(c) &\le \frac{F(u) - F(c)}{u-c} \le \frac{F(c)}{c} \le F'(0),\\ R'(c) &\le \frac{R(u) - 1}{u-c} \le \frac{1}{c} \le R'(0), \end{aligned} $$

(3.6)

for all $u \in (c,0)$. Note that by (P1′), either $F$ or $R$ is strictly convex, so that strict inequalities must hold in either the first or the second line. If $R'(0) < 0$, then applying the strictly increasing transformation $x \mapsto -\frac{1}{x}$ to the second line in (3.6) and multiplying term by term with the first, we obtain

$$ -\frac{F'(c)}{R'(c)} < -\frac{F(u) - F(c)}{R(u)-1} < -\frac{F(c)}{c} < -\frac{F'(0)}{R'(0)}. $$

Applying the integral $\frac{1}{c}\int _{0}^{c} \,du$ to all terms, (2.1) follows. If $R'(0) \ge 0$, this approach is still valid for the first two inequalities in each line of (3.6), but not for the last one. However, in the case $R'(0) \ge 0$, we have set $b_{\mathrm{inv}}= +\infty$ in (3.2), and the last inequality in (2.1) holds trivially. It remains to show that $D \cap (b_{\mathrm{y}\text{-norm}}, b_{\mathrm{inv}})$ is nonempty. $F$ is a convex function and by Condition 1.1 finite at least on the interval $(c,0)$. It follows that $F'(0) > -\infty$ and thus that $b_{\mathrm{inv}}> -\infty$ in general. If $D = {[0,\infty )}$, then $F'(0) > 0$ by (P3) and hence $b_{\mathrm{inv}}> 0$. Moreover $b_{\mathrm{y}\text{-norm}}< b_{\mathrm{asymp}}= -F(c) < \infty$, completing the proof. □