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Duffie-Kan Affine Term Structure Models

Duffie-Kan Affine Term Structure Models — Visual proof with formal theorem. Advanced Advanced Stochastic Processes at NICEFA.

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The Formal Theorem

Let the state vector Xt∈Rn X_t \in \mathbb{R}^n follow a Markov diffusion process under the risk-neutral measure Q \mathbb{Q} governed by the stochastic differential equation dXt=κ(θ−Xt)dt+Σ(Xt)dWt dX_t = \kappa(\theta - X_t)dt + \Sigma(X_t)dW_t . If the short rate rt r_t is an affine function of the state rt=δ0+δ1⋅Xt r_t = \delta_0 + \delta_1 \cdot X_t , then the price of a zero-coupon bond P(t,T) P(t, T) maturing at T T takes the exponential affine form:
P(t,T)=exp⁡(A(t,T)−B(t,T)⋅Xt) P(t, T) = \exp\left( A(t, T) - B(t, T) \cdot X_t \right)
where the deterministic functions A(t,T) A(t, T) and B(t,T) B(t, T) satisfy the associated system of ordinary differential equations (Riccati equations).

Analytical Intuition.

Imagine the yield curve as a living, breathing landscape of mountain ranges, where the state vector Xt X_t dictates the current elevation of interest rates. In the Duffie-Kan framework, we treat the bond market not as a chaotic sea of random noise, but as a structured, tractable geometry. Because the short rate rt r_t is tied linearly to Xt X_t , the expectation of the discount factor Et[exp⁡(−∫tTrsds)] \mathbb{E}_t [\exp(-\int_t^T r_s ds)] preserves this linear backbone. The exponential form is the 'DNA' of the model; it ensures that even as the state vector wanders through the multidimensional space of macroeconomic factors, the resulting bond prices remain analytically elegant. We are essentially mapping a complex stochastic path onto a static, predictable manifold. When we solve the Riccati equations, we are solving for the 'curvature' and 'tilt' of the yield curve, allowing us to decompose the future into a perfect, closed-form summation of risks. It turns the impossible task of path-dependent integration into a sleek, solvable problem of differential geometry.
CAUTION

Institutional Warning.

Students frequently conflate the state vector Xt X_t with the yield itself. Remember that Xt X_t represents latent factors (like inflation or growth), while the yield is a derived, global property of the entire term structure determined by the affine coefficients.

Academic Inquiries.

01

Why is the 'affine' property so significant for bond pricing?

The affine property guarantees that the term structure can be solved in closed form (up to ODEs), avoiding computationally expensive Monte Carlo simulations.

02

What role does the Riccati equation play?

It governs the evolution of the factor loadings B(t,T) B(t, T) , determining how sensitive a bond of maturity T T is to changes in the current state Xt X_t .

Standardized References.

  • Definitive Institutional SourceDuffie, D., & Kan, R. (1996). A Yield-Factor Model of Interest Rates.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Duffie-Kan Affine Term Structure Models: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/advanced-stochastic-processes/duffie-kan-affine-term-structure-models

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