Consider a short rate process (left(r_{t} ight)_{t in mathbb{R}_{+}})of the form (r_{t}=h(t)+X_{t}), where (h(t)) is a deterministic

Consider a short rate process \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\)of the form \(r_{t}=h(t)+X_{t}\), where \(h(t)\) is a deterministic function of time and \(\left(X_{t}\right)_{\mathbb{R}_{+}}\)is a Vasicek process started at \(X_{0}=0\).

a) Compute the price \(P(0, T)\) at time \(t=0\) of a bond with maturity \(T\), using \(h(t)\) and the function \(A(T)\) defined in (17.34) for the pricing of Vasicek bonds.

b) Show how the function \(h(t)\) can be estimated from the market data of the initial instantaneous forward rate curve \(f(0, t)\).