We consider a bond with maturity (T), priced (P(t, T)=) (mathbb{E}^{*}left[mathrm{e}^{-int_{t}^{T} r_{s} d s} mid mathcal{F}_{t} ight])

We consider a bond with maturity \(T\), priced \(P(t, T)=\) \(\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s} \mid \mathcal{F}_{t}\right]\) at time \(t \in[0, T]\).

a) Using the forward measure \(\widehat{\mathbb{P}}\) with numéraire \(N_{t}=P(t, T)\), apply the change of numéraire formula (16.9) to compute the derivative \(\frac{\partial P}{\partial T}(t, T)\).

b) Using Relation (18.5), find an expression of the instantaneous forward rate \(f(t, T)\) using the short rate \(r_{T}\) and the forward expectation \(\widehat{\mathbb{E}}\).

c) Show that the instantaneous forward rate \((f(t, T))_{t \in[0, T]}\) is a martingale under the forward measure \(\widehat{\mathbb{P}}\).