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Zero-Coupon Bond Pricing in the Vasicek Framework

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The Formal Theorem

Under the Vasicek model, the short rate rt r_t follows the SDE:
drt=a(b−rt)dt+σdWt dr_t = a(b - r_t)dt + \sigma dW_t
where a a is the speed of mean reversion, b b is the long-term mean, and σ \sigma is the volatility. The price P(t,T) P(t, T) of a zero-coupon bond maturing at T T is given by:
P(t,T)=A(t,T)e−B(t,T)rt P(t, T) = A(t, T)e^{-B(t, T)r_t}
where the deterministic functions are defined as:
B(t,T)=1−e−a(T−t)a B(t, T) = \frac{1 - e^{-a(T-t)}}{a}
A(t,T)=exp⁡((b−σ22a2)(B(t,T)−(T−t))−σ2B(t,T)24a) A(t, T) = \exp \left( (b - \frac{\sigma^2}{2a^2})(B(t, T) - (T-t)) - \frac{\sigma^2 B(t, T)^2}{4a} \right)

Analytical Intuition.

Imagine the short-term interest rate rt r_t as a tethered balloon in a turbulent sky. The parameter a a represents the strength of the elastic cord pulling the balloon toward the mean level b b , while σ \sigma describes the unpredictable gusts of wind (Brownian motion). Because the short rate is tied to this mean-reverting anchor, we can calculate the fair value of a bond by integrating the expected path of the short rate over the bond's life. The price formula decomposes this into two parts: B(t,T) B(t, T) captures the sensitivity of the bond price to current interest rate shocks—essentially acting as the 'duration'—while A(t,T) A(t, T) acts as a correction factor accounting for the volatility of the rate and the pull of the mean. Together, they transform our stochastic uncertainty into a deterministic price today, ensuring that our bond is priced consistently with the inherent physical 'gravity' of the market's mean-reversion dynamics.
CAUTION

Institutional Warning.

Students often struggle with the distinction between the physical measure and the risk-neutral measure. In the Vasicek framework, we implicitly assume the drift parameter a(b−rt) a(b-r_t) is already adjusted for market price of risk, meaning b b here represents the risk-neutral long-term mean, not necessarily the historical average.

Academic Inquiries.

01

Why is the Vasicek model preferred over a simple random walk for interest rates?

A random walk allows interest rates to drift to infinity, which is economically unrealistic. The Vasicek model incorporates mean reversion, forcing rates back toward a stable level, consistent with central bank behavior.

02

Can the Vasicek model produce negative interest rates?

Yes. Because the short rate follows a normal distribution N(E[rT],Var[rT]) N(E[r_T], Var[r_T]) , there is always a non-zero probability that rt r_t becomes negative, which is a known limitation of this model.

Standardized References.

  • Definitive Institutional SourceBrace, A., 'The Mathematics of Interest Rate Derivatives', Springer Finance.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Zero-Coupon Bond Pricing in the Vasicek Framework: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/advanced-stochastic-processes/zero-coupon-bond-pricing-in-the-vasicek-framework

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