Multi-Factor Term Structure Models

One-factor short-rate models have a fundamental structural deficiency: every point on the yield curve moves in perfect lockstep, driven by a single source of randomness. Empirical data contradicts this — principal component analysis of yield curve changes consistently shows that two or three factors are needed to explain 85–99% of the variance, corresponding to level, slope, and curvature movements that are far from perfectly correlated. A swaption depends jointly on rates at different tenors, and its correct pricing requires a model that can express the less-than-perfect correlation between, say, the 2-year and 10-year rate.