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1. Bond Affine Ansatz Models

Welcome to the affine term structure models section! This area explores the powerful class of interest rate models where bond prices take an exponential-affine form in the state variables — leading to closed-form pricing formulas and analytical tractability.

1.1 What Are Affine Models?

In affine term structure models, zero-coupon bond prices have the form:

\[ P(t, T) = A(t, T) \exp(-B(t, T) \cdot X(t)) \]

where \(X(t)\) represents the state variables (typically including the short rate) and \(A(t, T)\), \(B(t, T)\) are deterministic functions determined by the model dynamics.

This exponential-affine structure is incredibly powerful because:

  • Analytical solutions: We can derive closed-form bond prices without numerical methods
  • Computational efficiency: Fast pricing enables real-time risk management and calibration
  • Transparency: The roles of different parameters are clear and interpretable
  • Derivatives pricing: Many interest rate derivatives also admit closed-form or semi-analytical solutions

1.2 The Affine Ansatz Approach

The affine ansatz is a solution technique where we:

  1. Assume bond prices take an exponential-affine form
  2. Substitute this form into the fundamental pricing PDE
  3. Separate terms to obtain ODEs for \(A(t, T)\) and \(B(t, T)\)
  4. Solve these ODEs with appropriate boundary conditions

This approach transforms a stochastic pricing problem into a deterministic one, which is why affine models are so valuable in practice.

1.3 Models Covered

  • Vasicek Model — The classic Gaussian short rate model with mean reversion and closed-form bond pricing

  • CIR Model — Square-root diffusion ensuring non-negative rates, with Riccati ODEs leading to closed-form solutions

  • Hull-White Model — Time-dependent drift calibrated to market curves, maintaining Vasicek's analytical tractability

More extensions will be added, including multi-factor models, CIR++, and affine jump-diffusion frameworks.

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