Multi-Factor Affine Consistency and Exponential-Polynomial Families

Central banks worldwide estimate the yield curve every day using Nelson-Siegel or Svensson models — yet neither model is consistent with any non-trivial stochastic dynamics. There is no SDE that propagates a Nelson-Siegel curve forward in time without breaking the shape constraint. This is not a limitation of a particular parameterization; it is a theorem. The same analysis that explains why affine models (Vasicek, CIR, G2++) generate closed-form bond prices also explains the limits on what curve families can be self-consistent: the forward curve parameterization and the state variable SDE must jointly satisfy a PDE called the consistency condition, and most commonly used shapes fail it. This lesson derives the consistency condition, characterises the affine and polynomial term structure families, and shows that Nelson-Siegel is theoretically incompatible with arbitrage-free dynamics.