Vasicek Affine Term Structure Zero-Coupon Bond Pricing

Scenario

In the Vasicek short rate model, the instantaneous risk-free interest rate $r_t$ follows an Ornstein-Uhlenbeck mean-reverting process under the risk-neutral measure $\mathbb{Q}$: $dr_t = a(b - r_t)dt + \sigma dW_t^{\mathbb{Q}}$ where $a > 0$ is the speed of mean reversion, $b > 0$ is the long-term mean level, and $\sigma > 0$ is the rate volatility.

Your Task

1. Formulate the term structure partial differential equation for the zero-coupon bond price $P(t, T)$ with terminal boundary condition $P(T, T) = 1$.
2. Using the affine ansatz $P(t, T) = \exp(A(t, T) - B(t, T) r_t)$, derive the ordinary differential equations for $B(t, T)$ and $A(t, T)$.
3. State the closed-form expressions for $B(\tau)$ and $A(\tau)$ as functions of time-to-maturity $\tau = T - t$.
4. Compute the price and yield of a 5-year zero-coupon bond given: $r_0 = 0.03, a = 0.25, b = 0.05, \sigma = 0.02, T = 5.0$.

Constraints

• Constant model parameters $a, b, \sigma$.
• Standard affine term structure conditions.

Examples When $\tau \to 0$, $B(\tau) \approx \tau$ and $P \to e^{-r_0 \tau}$. As $\tau \to \infty$, the yield converges to an asymptotic constant level.