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

The Vasicek Model: Mean Reversion, Affine Bond Prices, and the Gaussian Distribution

A callable corporate bond's hedging sensitivity to long rates is very different from its sensitivity to short rates — a fact that the parallel-shift models of the Ho-Lee family cannot capture in an economically principled way. The Vasicek model (1977) addresses this by introducing mean reversion: when the short rate strays above its long-run value θ\theta, a restoring force pulls it back at rate kk. This single structural assumption changes the entire shape of the term structure — long rates approach θ\theta rather than extrapolating the short rate, and the volatility of long yields is dampened relative to short yields. The cost is accepting that the model cannot exactly fit today's curve; the gain is a consistent economic view of how rates behave over time.
Learning objectives. By the end of this note you should be able to:
  • write down the Vasicek SDE under Q\mathbb{Q} and identify kk, θ\theta, σ\sigma geometrically;
  • integrate the SDE explicitly and state the conditional distribution of r(t)r(t) given r(s)r(s);
  • derive the steady-state distribution and explain why mean reversion dampens long-yield volatility;
  • state the affine bond price formula P(t,T)=A(t,T)e−B(t,T)r(t)P(t,T) = A(t,T)e^{-B(t,T)r(t)} with explicit AA and BB;
  • compute a spot rate curve from Vasicek parameters and identify the role of each parameter;
  • articulate the negative-rate problem and why it is a tractability-versus-realism trade-off.

In this lesson

    1. The Ornstein-Uhlenbeck short-rate process
    1. Steady-state distribution and the term structure of volatility
    1. Affine bond price formula
    1. Negative rates: the tractability-versus-realism trade-off
    1. The Vasicek model as an equilibrium model
    1. Bond option pricing under Vasicek

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