Exercise . (A Gaussian HJM) Assume that the risk-neutral forward rate dynamics is given by df T t = ˆα (T − t) dt + β(T − t) dzQ t ,  ≤ t ≤ T, where zQ is a standard Brownian motion under Q, and

β(τ ) = ( + γ τ )σe

− ν

 τ

for non-negative constants σ, γ , and ν with γ >ν.

(a) Show that the forward rate volatility function β(τ )is humped, that is a τ ∗ > 

exists so that β is increasing for τ<τ ∗ and decreasing for τ>τ ∗.

(b) Compute the risk-neutral drift α(τ ) ˆ .

(c) What is the price of a European call option on a zero-coupon bond under the assumptions of this model?