Overview — Short-Rate Models

This topic is the engine room of fixed-income derivatives pricing. Lessons 1–2 establish the discrete framework — rate and price trees, no-arbitrage backwards induction, and the three forces (expectations, convexity, risk premium) that determine the shape of the term structure. Lessons 3–9 tour the canonical model zoo: from the simplest normally distributed models through mean-reverting Gaussian (Vasicek) and square-root (CIR) specifications to lognormal models (BDT, Black-Karasinski) and the calibrated extensions . Lesson 10 unifies the Gaussian and square-root families under the affine framework, showing that bond prices always reduce to an exponential-affine form $e^{A(\tau) - B(\tau)\,r(t)}$ whenever the model's drift and variance are affine in $r$. Lesson 11 closes the loop: given a short-rate model, how well does it fit the observed cap volatility curve and swaption matrix, and what does a real-market calibration look like?

By the end of this topic you should be able to:

• construct rate and price trees for a given short-rate specification, and price any rate derivative by no-arbitrage backwards induction;
• decompose the term structure slope into contributions from the expected short-rate path, Jensen's convexity correction, and a risk premium;
• calibrate a time-dependent drift (Ho-Lee) to exactly reproduce today's term structure;
• state the Vasicek and CIR SDEs, derive their steady-state distributions, and quote the closed-form bond price $P(t,T)$ in each case;
• derive the Vasicek bond price from the affine Riccati ODEs for $A(\tau)$ and $B(\tau)$;
• apply the Feller condition to CIR and explain why it prevents $r(t)$ from reaching zero;
• describe lognormal short-rate models (Dothan, BDT, Salomon, Black-Karasinski) and explain the skew consequences of a volatility proportional to $r$;
• extend Vasicek to Hull-White by deriving the deterministic shift $\varphi(t)$ that fits the initial forward curve, and build the corresponding trinomial tree;
• define the affine consistency condition and verify that Vasicek and CIR both satisfy it;
• compare models on their fit to the cap volatility term structure and swaption surface, and carry out a simple calibration to market data.

1. Contents
2. The big picture