Mean-Reverting Diffusions: OU, Vasicek, CIR, and the Feller Condition

Interest rate models in fixed income all belong to the same family: mean-reverting diffusions. The short rate is modelled as a process that is pulled back toward a long-run mean $\theta$ at rate $\kappa$, with randomness added by a Brownian shock. The two workhorses are:

The explicit solution for OU/Vasicek via the integrating factor $e^{\kappa t}$ is the cleanest example of an exactly solvable SDE. The CIR process is the square of a Bessel process (from Local Martingales and the Bessel Process); its transition density is non-central chi-squared, giving an explicit bond-pricing formula via the Feynman-Kac PDE.

In the Black-Scholes world, the short rate $r$ is constant. Switching to Vasicek or CIR introduces rate risk into bond prices, allowing the term structure (yield curve) to be modelled dynamically. Zero-coupon bond prices $P(t,T) = \mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r_s \, ds}|\mathcal{F}_t]$ become functions of $r_t$ alone (Markov property), and the Feynman-Kac PDE gives closed-form affine expressions $P(t,T) = A(t,T)e^{-B(t,T)r_t}$.

Learning objectives. By the end of this note you should be able to: