How do you derive analytical bond prices under the Vasicek model?

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AcadiFi · Accepted Answer

Vasicek assumes the short rate follows a mean-reverting Ornstein-Uhlenbeck process with constant volatility.

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The bond price takes the affine form P(t,T) = A(t,T) * exp(-B(t,T) * r_t), where B(t,T) = (1 - exp(-a(T-t))) / a and A(t,T) is a deterministic function of a, b, sigma. This affine structure is what makes Vasicek beautiful pedagogically.

Key properties:
- Yields are linear in r, so the term structure can slope up, down, or hump
- As T approaches infinity, yields converge to a long-run level depending on b, sigma, lambda (market price of risk)
- Rates can go NEGATIVE because r is Gaussian — the model's most criticized feature

For example, at Novara Asset Management, analysts initially used Vasicek for pedagogical modeling of 30-year treasuries, but switched to CIR when persistent near-zero rates in 2020 made Gaussian negative-rate scenarios uncomfortable (though post-2022 the negative-rate feature became arguably realistic again).

The closed-form price means you can instantly compute any zero-coupon bond or European bond option without Monte Carlo — a huge computational advantage for affine models.

How do you derive analytical bond prices under the Vasicek model?

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