Short-Rate Interest Rate Models › Short-Rate Interest Rate Models

In Part II, we studied the term structure of interest rates and learned how to extract discount factors and forward rates from observed bond prices. That analysis treated the yield curve as a static object, a snapshot of rates at a single point in time. But interest rates change constantly. The 10-year Treasury yield that was 4.2% yesterday might be 4.3% today and 3.9% next month. To price interest rate derivatives, manage bond portfolio risk, or forecast future rate scenarios, we need models that capture how interest rates evolve through time.

Short-rate models address this need by specifying a stochastic process for the instantaneous interest rate. Building on the Brownian motion and stochastic calculus framework from earlier chapters, these models describe rate dynamics with a combination of deterministic drift and random shocks. The key insight of short-rate models is that once you specify how the instantaneous rate evolves, the entire yield curve at any future time follows from no-arbitrage arguments.

This chapter introduces the two foundational short-rate models: the Vasicek model and the Cox-Ingersoll-Ross (CIR) model. Both incorporate mean reversion, the empirically observed tendency of interest rates to gravitate toward a long-run average level. We'll derive their dynamics, understand their parameters, simulate rate paths, and see how they produce closed-form bond prices. These models laid the groundwork for all subsequent interest rate modeling and remain essential tools for understanding rate dynamics.