Exercise 15.5 (HullWhite) Suppose that the spot-rate process {r(t)} under the risk-neutral probability measure Q follows the

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Exercise 15.5 (Hull–White) Suppose that the spot-rate process {r(t)} under the risk-neutral probability measure Q follows the SDE

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where a and σ are positive constants, ϕ(t) is a deterministic function of time t, and {z∗(t)} is a standard Brownian motion under Q. This affine model is called the Hull–White model (1990) or the extended Vasicek model. Show that the discount-bond price is given by

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Also, obtain the forward rate f(t, T) and its derivative with respect to t so as to prove that

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