Vasicek and CIR as Affine

The Vasicek and Cox-Ingersoll-Ross (CIR) models are the two canonical one-factor short-rate models, and both are affine processes. They illustrate the two fundamental cases of the affine framework: constant diffusion (Gaussian dynamics, Vasicek) and state-dependent diffusion (square-root dynamics, CIR). This section embeds both models in the affine framework, identifies the functions \(F\) and \(R\), derives the bond pricing Riccati solutions, and contrasts the two models systematically.

Learning Objectives

By the end of this section, you will be able to:

1. Write the Vasicek and CIR models in affine parameter form
2. Identify \(F\), \(R\), and the extended Riccati system for each model
3. Derive closed-form bond prices for both models
4. Compare Gaussian vs square-root dynamics and their financial implications

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Intuition

Both the Vasicek and CIR models describe a short rate \(r_t\) that mean-reverts toward a long-term level \(\theta\) with speed \(\kappa\). They differ in how volatility depends on the level:

• Vasicek: \(\sigma(r) = \sigma\) (constant). The rate can go negative. The transition density is Gaussian.
• CIR: \(\sigma(r) = \xi\sqrt{r}\) (proportional to \(\sqrt{r}\)). The rate stays non-negative (under the Feller condition). The transition density is non-central chi-squared.

In the affine classification, Vasicek is the \(A_0(1)\) model (zero CIR-type components among one total) and CIR is the \(A_1(1)\) model (one CIR-type component among one total). Despite their different volatility structures, both produce exponential-affine bond prices---the hallmark of affine term structure models.

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Vasicek Model

Dynamics and Affine Parameters

The Vasicek model under the risk-neutral measure:

\[ dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t \]

State space: \(D = \mathbb{R}\) (the rate is unconstrained), so \(m = 0\), \(d = 1\).

Affine parameters:

Parameter	Value	Interpretation
\(b_0\)	\(\kappa\theta\)	Constant drift
\(b_1\)	\(-\kappa\)	Mean-reversion speed
\(a_0\)	\(\sigma^2\)	Constant diffusion
\(a_1\)	\(0\)	No state-dependent volatility
\(\rho_0\)	\(0\)	Short rate offset
\(\rho_1\)	\(1\)	Short rate loading

Functions F and R

\[ F(w) = \kappa\theta\,w + \frac{1}{2}\sigma^2 w^2 \]

\[ R(w) = -\kappa\,w \]

Since \(a_1 = 0\), the function \(R\) is linear---the Riccati equation degenerates to a first-order linear ODE.

Bond Pricing Riccati System

With \(\rho_0 = 0\) and \(\rho_1 = 1\), the extended Riccati for bond pricing is:

\[ B'(\tau) = R(B(\tau)) - 1 = -\kappa B(\tau) - 1, \qquad B(0) = 0 \]

\[ A'(\tau) = F(B(\tau)) = \kappa\theta\,B(\tau) + \frac{1}{2}\sigma^2 B(\tau)^2, \qquad A(0) = 0 \]

Solution

The \(B\)-equation is a linear ODE with constant coefficients. The solution is:

\[ B(\tau) = -\frac{1 - e^{-\kappa\tau}}{\kappa} \]

Verification: \(B'(\tau) = -e^{-\kappa\tau}\) and \(-\kappa B(\tau) - 1 = (1 - e^{-\kappa\tau}) - 1 = -e^{-\kappa\tau}\). \(\square\)

For the \(A\)-equation, substituting \(B(\tau)\):

\[ A(\tau) = \left(\frac{\sigma^2}{2\kappa^2} - \theta\right)\!\left(B(\tau) + \tau\right) - \frac{\sigma^2}{4\kappa}B(\tau)^2 \]

Bond Price and Yield

\[ P(t, T) = \exp\!\left(A(\tau) + B(\tau)\,r_t\right) \]

The yield \(y(\tau) = -[A(\tau) + B(\tau)\,r_t]/\tau\) is linear in \(r_t\):

\[ y(\tau) = -\frac{A(\tau)}{\tau} + \frac{1 - e^{-\kappa\tau}}{\kappa\tau}\,r_t \]

As \(\tau \to \infty\), the yield converges to \(y(\infty) = \theta - \frac{\sigma^2}{2\kappa^2}\), the long rate.

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CIR Model

Dynamics and Affine Parameters

The CIR model under the risk-neutral measure:

\[ dr_t = \kappa(\theta - r_t)\,dt + \xi\sqrt{r_t}\,dW_t \]

State space: \(D = \mathbb{R}_+\) (the rate is non-negative), so \(m = 1\), \(d = 1\).

Affine parameters:

Parameter	Value	Interpretation
\(b_0\)	\(\kappa\theta\)	Constant drift
\(b_1\)	\(-\kappa\)	Mean-reversion speed
\(a_0\)	\(0\)	No constant diffusion
\(a_1\)	\(\xi^2\)	State-dependent volatility
\(\rho_0\)	\(0\)	Short rate offset
\(\rho_1\)	\(1\)	Short rate loading

Functions F and R

\[ F(w) = \kappa\theta\,w \]

\[ R(w) = -\kappa\,w + \frac{1}{2}\xi^2 w^2 \]

Since \(a_1 = \xi^2 \neq 0\), the function \(R\) is quadratic---a genuine Riccati equation.

Feller Condition

The CIR process stays strictly positive if and only if the Feller condition holds:

\[ 2\kappa\theta \geq \xi^2 \]

When violated, the process can reach zero but is instantaneously reflected. This condition translates to the admissibility requirement \((b_0)_1 = \kappa\theta > 0\) being strong enough relative to the diffusion.

Bond Pricing Riccati System

\[ B'(\tau) = -\kappa B(\tau) + \frac{1}{2}\xi^2 B(\tau)^2 - 1, \qquad B(0) = 0 \]

\[ A'(\tau) = \kappa\theta\,B(\tau), \qquad A(0) = 0 \]

Solution

Define the discriminant \(\gamma = \sqrt{\kappa^2 + 2\xi^2}\).

The \(B\)-equation is a scalar Riccati with constant term \(-1\), linear term \(-\kappa\), and quadratic coefficient \(\frac{1}{2}\xi^2\). By the substitution \(B = -\frac{2}{\xi^2}\frac{h'}{h}\) or by direct integration:

\[ B(\tau) = \frac{-2(e^{\gamma\tau} - 1)}{(\gamma + \kappa)(e^{\gamma\tau} - 1) + 2\gamma} \]

Verification: Computing \(B'(\tau)\) by the quotient rule and checking that \(B' = -\kappa B + \frac{1}{2}\xi^2 B^2 - 1\) is algebraically tedious but routine. \(\square\)

For the \(A\)-equation:

\[ A(\tau) = \frac{2\kappa\theta}{\xi^2}\log\!\left(\frac{2\gamma\,e^{(\gamma + \kappa)\tau/2}}{(\gamma + \kappa)(e^{\gamma\tau} - 1) + 2\gamma}\right) \]

Bond Price and Yield

\[ P(t, T) = \exp\!\left(A(\tau) + B(\tau)\,r_t\right) \]

Since \(B(\tau) < 0\), the bond price is decreasing in \(r_t\). The yield is:

\[ y(\tau) = -\frac{A(\tau)}{\tau} + \frac{2(e^{\gamma\tau} - 1)}{[(\gamma+\kappa)(e^{\gamma\tau}-1)+2\gamma]\tau}\,r_t \]

As \(\tau \to \infty\):

\[ y(\infty) = \frac{2\kappa\theta}{\gamma + \kappa} \]

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Side-by-Side Comparison

Feature	Vasicek	CIR
State space	\(\mathbb{R}\)	\(\mathbb{R}_+\)
Classification	\(A_0(1)\)	\(A_1(1)\)
Diffusion	\(\sigma\) (constant)	\(\xi\sqrt{r}\) (state-dependent)
\(R(w)\)	\(-\kappa w\) (linear)	\(-\kappa w + \frac{1}{2}\xi^2 w^2\) (quadratic)
\(B(\tau)\) ODE	Linear	Riccati
\(B(\tau)\) solution	\(-(1-e^{-\kappa\tau})/\kappa\)	Ratio of exponentials
Transition density	Gaussian	Non-central \(\chi^2\)
Negative rates	Possible	Impossible (Feller)
Long rate	\(\theta - \sigma^2/(2\kappa^2)\)	\(2\kappa\theta/(\gamma + \kappa)\)

Numerical Comparison

With \(\kappa = 0.5\), \(\theta = 0.05\), \(\sigma = \xi = 0.1\), \(r_0 = 0.03\):

Vasicek: \(B(5) = -(1 - e^{-2.5})/0.5 = -1.836\). Bond price \(P(0, 5) = e^{A(5) + B(5) \cdot 0.03}\).

CIR: \(\gamma = \sqrt{0.25 + 0.02} = \sqrt{0.27} = 0.5196\). \(B(5) = -2(e^{2.598} - 1)/[(1.0196)(e^{2.598} - 1) + 1.0392]\).

The CIR bond price is slightly higher because the square-root volatility reduces uncertainty when rates are low, effectively lowering the convexity adjustment. \(\square\)

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Summary

The Vasicek and CIR models are the simplest non-trivial affine term structure models. Vasicek, with constant diffusion (\(A_0(1)\)), produces a linear Riccati equation and Gaussian dynamics that permit negative rates. CIR, with square-root diffusion (\(A_1(1)\)), produces a genuine quadratic Riccati equation and non-central chi-squared dynamics that guarantee non-negative rates under the Feller condition. Both yield exponential-affine bond prices \(P(t, T) = e^{A(\tau) + B(\tau)r_t}\) with explicit solutions for \(A\) and \(B\), making them the building blocks of multi-factor affine term structure models.

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Further Reading

• Vasicek, O. (1977). "An Equilibrium Characterization of the Term Structure." Journal of Financial Economics, 5(2), 177-188.
• Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). "A Theory of the Term Structure of Interest Rates." Econometrica, 53(2), 385-407.
• Brigo, D. & Mercurio, F. Interest Rate Models - Theory and Practice. Springer, 2007, Chapters 3-4.

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Exercises

Exercise 1. For the Vasicek model, the bond price \(B(\tau)\) satisfies \(B'(\tau) = -1 - \kappa B(\tau)\) with \(B(0) = 0\). Solve this linear ODE to obtain \(B(\tau) = \frac{1}{\kappa}(e^{-\kappa\tau} - 1)\) and then integrate \(A'(\tau) = \kappa\theta B(\tau) + \frac{1}{2}\sigma^2 B(\tau)^2\) to derive the complete bond price formula.

Solution to Exercise 1

The ODE for \(B(\tau)\) is:

\[ B'(\tau) = -1 - \kappa B(\tau), \qquad B(0) = 0 \]

This is a first-order linear ODE. The integrating factor is \(e^{\kappa\tau}\). Multiplying both sides:

\[ \frac{d}{d\tau}\!\left[e^{\kappa\tau} B(\tau)\right] = -e^{\kappa\tau} \]

Integrating from \(0\) to \(\tau\):

\[ e^{\kappa\tau} B(\tau) - B(0) = -\frac{e^{\kappa\tau} - 1}{\kappa} \]

Since \(B(0) = 0\):

\[ B(\tau) = -\frac{1 - e^{-\kappa\tau}}{\kappa} = \frac{e^{-\kappa\tau} - 1}{\kappa} \]

Now substitute into the \(A\)-equation:

\[ A'(\tau) = \kappa\theta B(\tau) + \frac{1}{2}\sigma^2 B(\tau)^2 \]

Let \(\beta(\tau) = \frac{1 - e^{-\kappa\tau}}{\kappa}\) so that \(B(\tau) = -\beta(\tau)\). Then:

\[ A'(\tau) = -\kappa\theta\,\beta(\tau) + \frac{1}{2}\sigma^2\beta(\tau)^2 \]

Integrating \(\int_0^\tau \beta(s)\,ds = \frac{1}{\kappa}[\tau - \beta(\tau)]\) and \(\int_0^\tau \beta(s)^2\,ds\) by expanding and integrating term by term:

\[ A(\tau) = -\theta[\tau - \beta(\tau)] + \frac{\sigma^2}{2\kappa^2}\!\left[\tau - 2\beta(\tau) + \frac{1 - e^{-2\kappa\tau}}{2\kappa}\right] \]

Rearranging:

\[ A(\tau) = \left(\frac{\sigma^2}{2\kappa^2} - \theta\right)(B(\tau) + \tau) - \frac{\sigma^2}{4\kappa}B(\tau)^2 \]

The complete bond price is \(P(t, T) = \exp(A(\tau) + B(\tau)\,r_t)\) with \(\tau = T - t\).

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Exercise 2. Compare the long-run yield \(y_\infty = \lim_{\tau \to \infty} y(\tau)\) for the Vasicek and CIR models. For the Vasicek model, show that \(y_\infty = \theta - \frac{\sigma^2}{2\kappa^2}\), and for the CIR model, show that \(y_\infty = \frac{2\kappa\theta}{\gamma + \kappa}\) where \(\gamma = \sqrt{\kappa^2 + 2\xi^2}\). For which model can the long-run yield be negative?

Solution to Exercise 2

Vasicek long-run yield: As \(\tau \to \infty\), \(B(\tau) = -(1 - e^{-\kappa\tau})/\kappa \to -1/\kappa\). For the yield:

\[ y(\tau) = -\frac{A(\tau)}{\tau} - \frac{B(\tau)}{\tau}r_t \]

As \(\tau \to \infty\), \(B(\tau)/\tau \to 0\) (the \(r_t\)-dependent term vanishes). For \(A(\tau)/\tau\), using \(A(\tau) = (\frac{\sigma^2}{2\kappa^2} - \theta)(B(\tau) + \tau) - \frac{\sigma^2}{4\kappa}B(\tau)^2\):

\[ \frac{A(\tau)}{\tau} \to \frac{\sigma^2}{2\kappa^2} - \theta + 0 = \frac{\sigma^2}{2\kappa^2} - \theta \]

Therefore:

\[ y_\infty^{\text{Vas}} = -\left(\frac{\sigma^2}{2\kappa^2} - \theta\right) = \theta - \frac{\sigma^2}{2\kappa^2} \]

This can be negative when \(\sigma^2 > 2\kappa^2\theta\), i.e., when volatility is large relative to the mean-reversion speed and long-run level.

CIR long-run yield: As \(\tau \to \infty\), \(e^{\gamma\tau} \to \infty\), so:

\[ B(\tau) = \frac{-2(e^{\gamma\tau} - 1)}{(\gamma + \kappa)(e^{\gamma\tau} - 1) + 2\gamma} \to \frac{-2}{\gamma + \kappa} \]

\[ \frac{A(\tau)}{\tau} \to \frac{2\kappa\theta}{\xi^2}\cdot\frac{\gamma + \kappa}{2} - \frac{2\kappa\theta}{\xi^2}\cdot\frac{(\gamma + \kappa)}{2} + \ldots \]

More directly, using the known result:

\[ y_\infty^{\text{CIR}} = \frac{2\kappa\theta}{\gamma + \kappa} \]

where \(\gamma = \sqrt{\kappa^2 + 2\xi^2}\). Since \(\gamma > \kappa\), we have \(\gamma + \kappa > 2\kappa\), so \(y_\infty^{\text{CIR}} < \theta\). Moreover, since \(\kappa, \theta, \gamma > 0\), the CIR long-run yield is always positive: \(y_\infty^{\text{CIR}} > 0\).

Conclusion: Only the Vasicek model can produce a negative long-run yield (when \(\sigma^2/(2\kappa^2) > \theta\)). The CIR model always has \(y_\infty > 0\).

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Exercise 3. The Vasicek model has Gaussian transitions: \(r_T \mid r_t \sim N(\mu_\tau, \sigma_\tau^2)\). Derive the conditional mean \(\mu_\tau = \theta(1 - e^{-\kappa\tau}) + r_t e^{-\kappa\tau}\) and variance \(\sigma_\tau^2 = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa\tau})\) directly from the SDE. What is the stationary distribution as \(\tau \to \infty\)?

Solution to Exercise 3

The Vasicek SDE is \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\). This is a linear SDE (Ornstein-Uhlenbeck process). The solution by variation of constants is:

\[ r_T = e^{-\kappa\tau}r_t + \theta(1 - e^{-\kappa\tau}) + \sigma\int_t^T e^{-\kappa(T-s)}\,dW_s \]

Conditional mean: The stochastic integral has zero expectation, so:

\[ \mu_\tau = \mathbb{E}[r_T \mid r_t] = \theta(1 - e^{-\kappa\tau}) + r_t e^{-\kappa\tau} \]

This is a weighted average of \(r_t\) and \(\theta\), with the weight on \(r_t\) decaying exponentially.

Conditional variance: By Ito's isometry:

\[ \sigma_\tau^2 = \operatorname{Var}(r_T \mid r_t) = \sigma^2 \int_t^T e^{-2\kappa(T-s)}\,ds = \sigma^2 \cdot \frac{1 - e^{-2\kappa\tau}}{2\kappa} \]

Stationary distribution: As \(\tau \to \infty\):

\[ \mu_\infty = \theta, \qquad \sigma_\infty^2 = \frac{\sigma^2}{2\kappa} \]

The stationary distribution is \(N(\theta, \sigma^2/(2\kappa))\). The rate fluctuates around \(\theta\) with a standard deviation of \(\sigma/\sqrt{2\kappa}\), confirming that faster mean reversion (\(\kappa\) large) or lower volatility (\(\sigma\) small) concentrates the rate around \(\theta\).

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Exercise 4. For the CIR model, the transition density involves the non-central chi-squared distribution. Without deriving the density, use the characteristic function to compute the conditional mean \(\mathbb{E}[r_T \mid r_t]\) and variance \(\operatorname{Var}(r_T \mid r_t)\). Verify that \(\mathbb{E}[r_T \mid r_t] = \theta(1 - e^{-\kappa\tau}) + r_t e^{-\kappa\tau}\) (same mean as Vasicek) but the variance differs.

Solution to Exercise 4

The CIR SDE is \(dr_t = \kappa(\theta - r_t)\,dt + \xi\sqrt{r_t}\,dW_t\). The conditional mean can be computed directly by taking expectations of the SDE. Since \(\mathbb{E}[dW_t] = 0\):

\[ \frac{d}{dt}\mathbb{E}[r_t] = \kappa(\theta - \mathbb{E}[r_t]) \]

This linear ODE for \(m(t) = \mathbb{E}[r_t \mid r_0]\) has the solution:

\[ \mathbb{E}[r_T \mid r_t] = \theta(1 - e^{-\kappa\tau}) + r_t e^{-\kappa\tau} \]

This is identical to the Vasicek conditional mean.

For the variance, define \(v(t) = \operatorname{Var}(r_t \mid r_0)\). Using Ito's formula on \(r_t^2\):

\[ d(r_t^2) = 2r_t\,dr_t + (dr_t)^2 = 2r_t[\kappa(\theta - r_t)\,dt + \xi\sqrt{r_t}\,dW_t] + \xi^2 r_t\,dt \]

Taking expectations:

\[ \frac{d}{dt}\mathbb{E}[r_t^2] = 2\kappa\theta\,\mathbb{E}[r_t] - 2\kappa\,\mathbb{E}[r_t^2] + \xi^2\,\mathbb{E}[r_t] \]

Since \(v(t) = \mathbb{E}[r_t^2] - (\mathbb{E}[r_t])^2\), combining with the mean equation yields:

\[ v'(t) = -2\kappa\,v(t) + \xi^2\,m(t) \]

Substituting \(m(t) = \theta(1 - e^{-\kappa t}) + r_0 e^{-\kappa t}\) and solving the linear ODE with \(v(0) = 0\):

\[ \operatorname{Var}(r_T \mid r_t) = r_t \cdot \frac{\xi^2}{\kappa}(e^{-\kappa\tau} - e^{-2\kappa\tau}) + \theta \cdot \frac{\xi^2}{2\kappa}(1 - e^{-\kappa\tau})^2 \]

Unlike the Vasicek variance \(\sigma^2(1 - e^{-2\kappa\tau})/(2\kappa)\), the CIR variance depends on the current rate \(r_t\). When \(r_t\) is large, the variance is larger (because the diffusion \(\xi\sqrt{r_t}\) is larger). This state-dependent variance is the key difference from the Vasicek model.

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Exercise 5. The Feller condition for the CIR model is \(2\kappa\theta \geq \xi^2\). Interpret each side of this inequality financially: what does \(\kappa\theta\) represent, and what does \(\xi^2\) represent? Explain intuitively why the mean reversion "pull" must dominate the volatility for the rate to stay strictly positive.

Solution to Exercise 5

The Feller condition is \(2\kappa\theta \geq \xi^2\).

Left side (\(\kappa\theta\)): The quantity \(\kappa\theta\) is the "pull" of mean reversion at \(r = 0\). When \(r_t = 0\), the drift is \(\kappa(\theta - 0) = \kappa\theta > 0\), pushing the rate upward. The product \(\kappa\theta\) measures the strength of this restoring force: \(\kappa\) is the speed at which the rate is pulled back, and \(\theta\) is how far the target is from zero. Together, \(\kappa\theta\) is the instantaneous drift at the origin.

Right side (\(\xi^2\)): The quantity \(\xi^2\) measures the volatility's ability to push the rate toward zero. Near \(r = 0\), the diffusion coefficient is \(\xi\sqrt{r}\), and the quadratic variation is \(\xi^2 r\). The term \(\xi^2\) appears because of the Ito correction: \(d(\sqrt{r_t})\) involves a term \(-\xi^2/(8\sqrt{r_t})\,dt\) that pulls \(\sqrt{r_t}\) toward zero. The larger \(\xi^2\), the stronger this downward pressure.

Intuition: The rate stays strictly positive when the deterministic drift at the boundary (\(\kappa\theta\)) is strong enough to overcome the random fluctuations that push it toward zero (scaled by \(\xi^2/2\)). If \(2\kappa\theta < \xi^2\), the volatility-induced downward drift dominates near zero, and the process can reach the boundary. The factor of 2 arises from the Ito correction in the SDE for \(\sqrt{r_t}\).

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Exercise 6. Plot (or sketch) the yield curves \(y(\tau) = -\frac{A(\tau)}{\tau} - \frac{B(\tau)}{\tau}r_0\) for the Vasicek model with \(\kappa = 0.5\), \(\theta = 0.05\), \(\sigma = 0.02\) for three different starting rates: \(r_0 = 0.02\) (below \(\theta\)), \(r_0 = 0.05\) (at \(\theta\)), and \(r_0 = 0.08\) (above \(\theta\)). Describe the shape of each curve (normal, inverted, or humped).

Solution to Exercise 6

With \(\kappa = 0.5\), \(\theta = 0.05\), \(\sigma = 0.02\), the key quantities are:

\[ B(\tau) = -\frac{1 - e^{-0.5\tau}}{0.5} = -2(1 - e^{-0.5\tau}) \]

\[ y_\infty = \theta - \frac{\sigma^2}{2\kappa^2} = 0.05 - \frac{0.0004}{0.5} = 0.05 - 0.0008 = 0.0492 \]

The yield at maturity \(\tau\) is \(y(\tau) = -A(\tau)/\tau - B(\tau) r_0/\tau\). At \(\tau \to 0\), \(y(0^+) = r_0\) (the instantaneous short rate). As \(\tau \to \infty\), \(y \to 0.0492\).

Case \(r_0 = 0.02\) (below \(\theta\)): The yield starts at \(y(0^+) = 0.02\) and rises toward \(y_\infty = 0.0492\). The curve is normal (upward sloping), reflecting the expectation that rates will mean-revert upward from their currently low level.

Case \(r_0 = 0.05\) (at \(\theta\)): The yield starts at \(y(0^+) = 0.05\) and decreases slightly toward \(y_\infty = 0.0492\). The curve is nearly flat with a slight downward slope, reflecting only the convexity adjustment \(-\sigma^2/(2\kappa^2) = -0.0008\). This is a mildly inverted (slightly downward-sloping) curve.

Case \(r_0 = 0.08\) (above \(\theta\)): The yield starts at \(y(0^+) = 0.08\) and falls toward \(y_\infty = 0.0492\). The curve is inverted (downward sloping), reflecting the market's expectation that the currently high rate will mean-revert downward toward \(\theta = 0.05\).

In all three cases, the long end converges to the same value \(y_\infty = 0.0492\). The shape of the yield curve is determined by the position of \(r_0\) relative to \(\theta\): below \(\theta\) gives a normal curve, at \(\theta\) gives a flat curve (modulo convexity), and above \(\theta\) gives an inverted curve.