If we have a model for the short rate \(r(t)\), show that

(a) the zero coupon bond price can be calculated as:

\[P(t, s)=\mathbf{E}_{t}\left[e^{\int_{t}^{s} r(u) d u}\right]=\mathbf{E}\left[e^{\int_{t}^{s} r(u) d u} \mid \mathcal{F}_{t}\right]\]

(b) the price of an option that pays \(C(T)\) at time \(T\) is:

\[C(t)=\mathbf{E}_{t}\left[e^{\int_{t}^{s} r(u) d u} C(T)\right]=\mathbf{E}\left[\int^{\int_{t}^{s} r(u) d u} C(T) \mid \mathcal{F}_{t}\right]\]

where \(r(t)\) is the interest rate of a loan emitted at \(t\) with immediate expiration \(t+\Delta t\) as perceived now at time zero.