Affine Term-Structure Models: Unifying Vasicek and CIR

Every closed-form bond price derived in this unit — Vasicek, CIR, Ho-Lee, Hull-White — shares a single structural form: $P(t,T) = A(t,T)e^{-B(t,T)r(t)}$. This note asks why that form works and exactly which models admit it. The answer is the affine consistency theorem: a model admits closed-form $A/B$ bond prices if and only if its drift and squared diffusion are both affine (linear with an intercept) in the short rate. The theorem partitions the short-rate universe cleanly into two non-degenerate affine classes — Gaussian (Vasicek/Hull-White family) and square-root (CIR family) — and a non-affine wilderness (Dothan, BDT, Black-Karasinski). Knowing which class a model lives in tells practitioners immediately whether closed-form swaption pricing, yield-curve fitting, and option-adjusted spread analysis are within reach — or whether trees and Monte Carlo are unavoidable.