Short-Rate Interest Rate Models › Summary

This chapter introduced short-rate models, which describe the evolution of the instantaneous interest rate as a stochastic process with mean reversion. The key concepts covered include:

The short rate $r(t)$ is the instantaneous interest rate driving all bond prices through the relationship $P(t,T) = \mathbb{E}^Q[\exp(-\int_t^T r(s)\,ds)]$. Specifying a stochastic process for $r(t)$ determines the entire term structure dynamics.

The Vasicek model $dr = \kappa(\theta - r)dt + \sigma dW$ was the first mean-reverting short-rate model. It produces normally distributed rates, allowing for negative values, and has simple closed-form bond pricing formulas with exponential-affine structure.

The CIR model $dr = \kappa(\theta - r)dt + \sigma\sqrt{r}dW$ modifies Vasicek by making volatility proportional to $\sqrt{r}$. This keeps rates non-negative when the Feller condition $2\kappa\theta \geq \sigma^2$ holds, producing chi-squared distributed rates with state-dependent volatility.

Mean reversion is the central feature of both models. The parameter $\kappa$ controls how fast rates return to the long-run mean $\theta$ after a shock. Higher $\kappa$ means faster reversion and lower long-run variance.

Yield curves under these models converge to a common long-term rate regardless of the initial short rate. The shape depends on whether $r(0)$ is above or below $\theta$, producing inverted, flat, or normal curves.

Limitations include the single-factor structure (perfect correlation across maturities), inability to match observed yield curves exactly, and constant parameters that cannot capture regime changes.

These models laid the foundation for all modern interest rate modeling. The next chapter examines advanced frameworks like HJM and the LIBOR Market Model that address some limitations while preserving the arbitrage-free pricing foundation established here.