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Fixed Income and Interest-Rate Models / Short-Rate Models

Derivation — Vasicek Bond Price via the Term-Structure PDE and Affine Riccati ODEs

The Vasicek model promises a closed-form bond price P(t,T)=A(t,T)e−B(t,T)r(t)P(t,T) = A(t,T)e^{-B(t,T)r(t)}, but why does this formula exist and how are AA and BB computed? The answer is a two-step argument: (i) apply the Feynman-Kac theorem to convert the pricing expectation into a PDE, then (ii) show that an exponential-affine trial solution reduces the PDE to a pair of ordinary differential equations — one linear for B(τ)B(\tau), one integrable for A(τ)A(\tau). This machinery works for any affine short-rate model, making it the foundational calculation behind all of Vasicek, CIR, Ho-Lee, and Hull-White in a unified framework.
Learning objectives. By the end of this note you should be able to:
  • state the term-structure PDE and identify it as the Feynman-Kac equation for a diffusion short-rate model;
  • carry out the affine ansatz substitution and derive the ODE system for A(t,T)A(t,T) and B(t,T)B(t,T);
  • solve the linear ODE for B(τ)B(\tau) and integrate to get A(τ)A(\tau) for the Vasicek model;
  • state the affine consistency condition that characterises which short-rate SDEs admit exponential-affine bond prices;
  • derive the Vasicek forward rate formula and read off the convexity correction.

In this lesson

    1. The term-structure equation
    1. The affine ansatz and the ATS characterisation
    1. Solving the Vasicek ODEs
    1. The affine structure in context

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