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Vasicek Model: Pricing Zero-Coupon Bonds in a Stochastic Interest Rate World

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The Formal Theorem

Under the risk-neutral measure Q \mathbb{Q} , let the short rate rt r_t follow the Ornstein-Uhlenbeck process:
drt=a(b−rt)dt+σdWt dr_t = a(b - r_t)dt + \sigma dW_t
where a a is the speed of reversion, b b is the long-term mean, and σ \sigma is volatility. The price at time 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:
B(t,T)=1−e−a(T−t)a B(t, T) = \frac{1 - e^{-a(T-t)}}{a}
A(t,T)=exp⁡((B(t,T)−(T−t))(b−σ22a2)−σ2B(t,T)24a) A(t, T) = \exp\left( \left( B(t, T) - (T-t) \right) \left( b - \frac{\sigma^2}{2a^2} \right) - \frac{\sigma^2 B(t, T)^2}{4a} \right)

Analytical Intuition.

Imagine the interest rate as a tethered particle in a fluid of perpetual volatility. In the Vasicek model, this particle is not aimless; it is governed by a gravitational pull toward a central mean b b , mediated by the stiffness a a . If the rate spikes, the 'restoring force' drags it back; if it dips, the same force pulls it up. When we price a zero-coupon bond, we are effectively calculating the present value of a guaranteed 1 1 dollar at maturity T T . Because the rate rt r_t is stochastic, we must integrate the entire path of these fluctuations. Mathematically, this yields an affine term structure where the bond price is an exponential function of the current rate. The B(t,T) B(t, T) term captures the sensitivity (duration) of the bond to rate movements, while A(t,T) A(t, T) functions as a convexity adjustment that accounts for the 'jitter' of the short rate over time. It is the synthesis of mean reversion and uncertainty, mapping the chaotic dance of rates into a deterministic price.
CAUTION

Institutional Warning.

Students often mistake the Vasicek model for a Black-Scholes derivative. Crucially, Vasicek is an equilibrium model for the *short rate itself*, not an asset price. Furthermore, the model allows rt r_t to become negative, which is mathematically convenient but theoretically polarizing in interest rate modeling.

Academic Inquiries.

01

Why does the Vasicek model include a mean reversion parameter?

Economic theory suggests that interest rates cannot drift to infinity. The mean reversion parameter a a ensures the process is stationary, reflecting the central bank's control and macroeconomic equilibrium.

02

What is the primary limitation of the Vasicek model?

The model assumes constant volatility σ \sigma and does not inherently prevent negative interest rates, which contradicts empirical evidence during certain market regimes.

Standardized References.

  • Definitive Institutional SourceBrigo, D., & Mercurio, F., Interest Rate Models - Theory and Practice

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Vasicek Model: Pricing Zero-Coupon Bonds in a Stochastic Interest Rate World: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/vasicek-model--pricing-zero-coupon-bonds-in-a-stochastic-interest-rate-world

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