Under Vasicek, dr = kappa(theta - r) dt + sigma dW and the zero coupon bond price is P(0,T) = A(T) exp(-B(T) r_0). Derive B(T), then price the 5 year zero with kappa = 0.3, theta = 0.05, sigma = 0.01 and r_0 = 3%.

Under Vasicek, dr = kappa(theta - r) dt + sigma dW and the zero coupon bond price is P(0,T) = A(T) exp(-B(T) r_0). Derive B(T), then price the 5 year zero with kappa = 0.3, theta = 0.05, sigma = 0.01 and r_0 = 3%.

Approach: Substitute the exponential affine guess into the term structure PDE, match the coefficients of r to obtain an ordinary differential equation for B, then evaluate A from the remaining equation.

0.8208. Substituting the exponential affine guess into the Vasicek term structure PDE and matching the coefficient of r gives the Riccati equation B'(T) = 1 - kappa B(T) with B(0) = 0, so B(T) = (1 - e^{-kappa T})/kappa, and the remaining equation integrates to ln A(T) = (B(T) - T)(kappa^2 theta - sigma^2/2)/kappa^2 - sigma^2 B(T)^2/(4 kappa). This is the affine term structure form, linear in r inside the exponent, which is what makes the Vasicek model tractable. With kappa = 0.3 and T = 5, B = (1 - e^{-1.5})/0.3 = 2.5896, so B r_0 = 0.07769, and ln A = (2.5896 - 5)(0.09 * 0.05 - 0.00005)/0.09 - 0.0001(6.7059)/1.2 = -0.11918 - 0.00056 = -0.11974. The zero coupon bond price is exp(-0.11974 - 0.07769) = 0.8208, a continuously compounded yield of 3.95%.

Follow-up: What is the probability of a negative short rate at 5 years under these parameters, and what does that do to the price of a floor struck at zero?

Key concepts: affine term structure, Vasicek model, Riccati equation, zero coupon bond.