Vasicek and CIR short rate models

Definition A short-rate model specifies the dynamics of the instantaneous rate $r$ under a pricing measure $\mathbb{Q}$ for the money-market account $B_t = \exp\bigl(\int_0^t r_s \, ds\bigr)$, here as the solution of a time-homogeneous equation

that has a unique strong solution. Default-free zero-coupon bond prices then follow from the pricing formula with the account as numeraire. With the integrated rate $I_{t,T} = \int_t^T r_s \, ds$,

assuming the expectation is finite. Because $r$ is a Markov process, the conditional law of its future path given $\mathcal{F}_t$ depends only on $r_t$, so $P(t,T) = p(t, r_t)$ for a deterministic function $p$, and in a one-factor model the whole curve $T \mapsto P(t,T)$ is driven by the single number $r_t$. When $I_{t,T}$ is $\mathbb{Q}$-integrable, Jensen’s inequality for the strictly convex function $x \mapsto e^{-x}$ gives

with equality only when the integrated rate is already known at time $t$. So the expected rate path alone does not price the bond. Every parameter in this lesson belongs to $\mathbb{Q}$ unless it is stated otherwise.

Definition The Vasicek model is the short-rate model

with constants $\kappa > 0$ per year, $\theta \in \mathbb{R}$ and $\sigma \ge 0$ per square-root year, and a known $r_0$. Its drift has the mean-reversion form. The term $\kappa(\theta - r)$ pulls the rate toward the level $\theta$ with speed $\kappa$, and for $\tau = T - t$ the explicit solution is

Given $\mathcal{F}_t$, the rate $r_T$ is Gaussian with mean $\theta + (r_t - \theta)e^{-\kappa\tau}$ and variance $\sigma^2 (1 - e^{-2\kappa\tau})/(2\kappa)$. The expected distance to $\theta$ halves every $\ln 2/\kappa$ years, its half-life, and the stationary law is $N(\theta, \sigma^2/(2\kappa))$. With $B(\tau) = (1 - e^{-\kappa\tau})/\kappa$, a stochastic Fubini interchange writes the integrated rate as $\theta\tau + (r_t - \theta)B(\tau)$ plus $\sigma \int_t^T B(T - u)\,dW^{\mathbb{Q}}_u$, so given $\mathcal{F}_t$ it is Gaussian with that mean $M$ and, by the Itô isometry, the variance

The Gaussian exponential moment then gives the bond price $P(t,T) = e^{-M + V/2}$, finite on every finite horizon. When $\sigma > 0$ the rate is negative with positive probability at every future date. That is a property of the Gaussian model, and the bond price stays positive, although it can exceed $1$ when rates are expected to be negative.

Definition A short-rate model has an exponential-affine, or affine, bond price if

with $\tau = T - t$, for deterministic functions with $A(0) = B(0) = 0$. When the drift $\mu(r)$ and the squared diffusion $s(r)^2$ are affine in $r$, the Feynman-Kac equation

with $p(T, r) = 1$ has a solution of this form whenever the two equations below have solutions on the horizon, as they do for Vasicek and CIR. Substituting $\partial_t p = -(A' - B' r)p$, $\partial_r p = -Bp$ and $\partial_{rr} p = B^2 p$ and collecting the terms that multiply $r$ and the constant terms gives two ordinary differential equations. For Vasicek, with $\mu(r) = \kappa(\theta - r)$ and $s^2 = \sigma^2$, they are

with the solutions $B(\tau) = (1 - e^{-\kappa\tau})/\kappa$ and

where $B = B(\tau)$. Expanding $-M + V/2$ from the Gaussian route gives the same $A(\tau) - B(\tau) r_t$, which confirms that this solution of the equation is the expectation. Itô’s formula applied to the bond gives $dP/P = r_t \, dt - B(\tau)\, s(r_t)\, dW^{\mathbb{Q}}_t$, so $-B(\tau)$ is the sensitivity of the log price to the short rate, and in Vasicek the bond loading is $-\sigma B(\tau)$.

Definition The Cox-Ingersoll-Ross model, or CIR model, is the short-rate model

with constants $\kappa > 0$, $\theta > 0$ and $\xi > 0$ and a known $r_0 \ge 0$. This equation has a unique strong solution, and that solution stays nonnegative, a result stated here without proof. The drift is the same mean-reverting drift as in Vasicek, so the conditional mean of $r_T$ is again $\theta + (r_t - \theta)e^{-\kappa\tau}$, but the instantaneous variance $\xi^2 r_t$ is proportional to the level. The squared diffusion is affine in $r$, so the bond price is affine, and its equations are

with $A(0) = B(0) = 0$. The quadratic $1 - \kappa B - \tfrac12 \xi^2 B^2$ has the roots $(-\kappa \pm \gamma)/\xi^2$, and separating variables in this Riccati equation and integrating $A'$ gives, with

the solution

where $\mathcal{A} = e^{A}$, and the bond price $P(t,T) = \mathcal{A}(\tau)\, e^{-B(\tau) r_t}$. The exponent $2\kappa\theta/\xi^2$ needs $\xi > 0$. The zero-diffusion case is the limit $\xi \to 0$, which is deterministic discounting along the mean-reverting path, $P(t,T) = \exp\bigl(-\theta\tau - (r_t - \theta)(1 - e^{-\kappa\tau})/\kappa\bigr)$.

Definition For the CIR model with $\kappa, \theta, \xi > 0$, the standard boundary classification, stated here without proof, says the following for $r_0 > 0$.

1. If the Feller condition $2\kappa\theta \ge \xi^2$ holds, then $r_t > 0$ for all $t \ge 0$ almost surely, so zero is never reached.
2. If $2\kappa\theta < \xi^2$, the rate reaches zero with positive probability, and at zero the diffusion vanishes while the drift $\kappa\theta > 0$ pushes it straight back into positive values.

In both cases $r_t \ge 0$ for all $t$. Failing the Feller condition allows the value zero, never negative values. Started at $r_0 = 0$, the process leaves zero at once, and under the Feller condition it never returns. Equivalent measures agree on which events have probability zero, and reaching zero before a fixed horizon is such an event, so when the physical and the pricing dynamics are both CIR under equivalent measures, the Feller condition holds under both or under neither. The bond formula of the previous card does not use the Feller condition and holds in both cases. The Vasicek model is different in kind, since with $\sigma > 0$ its Gaussian rate is negative with positive probability at every future date.

Definition The parameters in a bond formula are those of the $\mathbb{Q}$-dynamics. Under the physical measure the same Vasicek rate may follow $dr_t = \kappa^{\mathbb{P}}(\theta^{\mathbb{P}} - r_t)\,dt + \sigma \, dW^{\mathbb{P}}_t$. With the course’s Girsanov convention

where the market price of risk $\lambda$ is predictable and the density is a true martingale, the process $W^{\mathbb{Q}}_t = W^{\mathbb{P}}_t + \int_0^t \lambda_s \, ds$ is a $\mathbb{Q}$-Brownian motion and the $\mathbb{Q}$-drift of $r$ is $\kappa^{\mathbb{P}}(\theta^{\mathbb{P}} - r_t) - \sigma\lambda_t$. The diffusion coefficient $\sigma$ is the same under both measures, since quadratic variation does not change under an equivalent change of measure, while the drift changes. The $\mathbb{Q}$-dynamics are again of Vasicek form only for a suitable specification, for example the affine choice $\lambda_t = \lambda_0 + \lambda_1 r_t$, which gives

with mean reversion under $\mathbb{Q}$ when $\kappa^{\mathbb{Q}} > 0$. A constant $\lambda$ keeps $\kappa$ and sets $\theta^{\mathbb{Q}} = \theta^{\mathbb{P}} - \sigma\lambda/\kappa^{\mathbb{P}}$. For CIR, the classical choice $\lambda_t = \bar{\lambda}\sqrt{r_t}$ keeps the square-root form with $\kappa^{\mathbb{Q}} = \kappa^{\mathbb{P}} + \xi\bar{\lambda}$ and the same product $\kappa\theta$, under conditions that make the density a true martingale. A time-series estimate of $\kappa^{\mathbb{P}}$ and $\theta^{\mathbb{P}}$ is therefore not a pricing calibration until the market price of risk is specified, and bond prices inform the $\mathbb{Q}$-parameters.