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

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

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

Transcribed Image Text:

dr = ((t)ar)dt+odz*, 0