Exercise . (The Hull–White model calibrated to the Vasicek yield curve) Suppose the observable bond prices are fitted to a discount function of the form

(

∗)B¯(t) = e

−a(t)−b(t)r , where b(t) = 

κ

 − e−κt



, a(t) = y∞[t − b(t)] + β

κ

b(t)



, where y∞, κ, and β are constants. This is the discount function of the Vasicek model, see (.)–(.).

(a) Express the initial forward rates ¯f(t) and the derivatives ¯f

(t)in terms of the functions a and b.

(b) Show by substitution into (.) that the function θ (ˆ t) in the Hull–White model will be given by the constant