Can you explain affine term structure models in plain language?

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AcadiFi · Accepted Answer

An affine term structure model assumes the short rate is a linear (affine) function of a small number of latent state variables, typically two or three factors capturing level, slope, and curvature. Yields of any maturity then also become affine functions of those same factors, which makes pricing every bond on the curve computationally cheap. The short rate follows r(t) = delta0 + delta1' * X(t), where X(t) is the state vector following a Gaussian or square-root diffusion. Bond prices take the form exp(A(tau) - B(tau)' * X(t)), where A and B solve ordinary differential equations derived from no-arbitrage conditions. ```mermaid flowchart LR A[State Factors
X(t)] --> B[Short Rate
r(t)] B --> C[Risk-Neutral
Dynamics] C --> D[Bond Prices
P(t,T)] D --> E[Yield Curve
y(t,T)] ``` ATSMs appear throughout risk management: term premium decomposition, scenario generation for ALM, and swaption pricing under no-arbitrage calibration. The tradeoff is that Gaussian models admit negative rates while CIR-type models stay positive but have heavier calibration burden. For the FRM exam, know that Vasicek and CIR are the simplest one-factor affine models.

Can you explain affine term structure models in plain language?

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