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 . 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 . Below, the correct mathematical expression for the threshold is given, supplemented with a self-contained and corrected proof.

## Setting

In , 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 . 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  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 , with the convention that $$\lambda = 0$$ if no negative solution $$c$$ exists. The second objective of  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. ), we require that

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

## Corrections to results

Theorem 3.1 in  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. (1)

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

2. (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  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  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. (1)

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

2. (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  it was erroneously claimed that $$b_{\mathrm{y}\text{-}\mathrm{norm}}= b_{\mathrm{fw} \text{-}\mathrm{norm}}$$.

Corollary 3.12 in  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)].

## 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. (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. (2)

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

3. (3)

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

4. (4)

Either

1. a)

$$D = {[0,\infty )}$$, or

2. 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. (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  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}

\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. □