In the Vasicek model, set f(t,r) = exp((T t) (T t)r) for fixed T. (a) Show that

In the Vasicek model, set f(t,r) = exp(−α(T −t) −β(T −t)r) for fixed T.

(a) Show that f satisfies the fundamental PDE and the boundary condition f(T,r) = 1 if and only if

β

+κβ = 1 ,

α

− κθβ +

1 2

σ2

β2 = 0 , and α(0) = β(0) = 0.

(b) Verify that the formula (18.18) for yields with β(τ ) = τ b(τ ) and

α(τ ) = τ a(τ ) satisfies these conditions.