We work in the short rate model [d r_{t}=sigma d B_{t}] where (left(B_{t} ight)_{t in mathbb{R}_{+}})is a

We work in the short rate model

\[d r_{t}=\sigma d B_{t}\]

where \(\left(B_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under \(\mathbb{P}^{*}\), and \(\widehat{\mathbb{P}}_{2}\) is the forward measure defined by

\[\frac{\mathrm{d} \widehat{\mathbb{P}}_{2}}{\mathrm{~d} \mathbb{P}^{*}}=\frac{1}{P\left(0, T_{2}\right)} \mathrm{e}^{-\int_{0}^{T_{2}} r_{s} d s}\]

a) State the expressions of \(\zeta_{1}(t)\) and \(\zeta_{2}(t)\) in

\[\frac{d P\left(t, T_{i}\right)}{P\left(t, T_{i}\right)}=r_{t} d t+\zeta_{i}(t) d B_{t}, \quad i=1,2\]

and the dynamics of the \(P\left(t, T_{1}\right) / P\left(t, T_{2}\right)\) under \(\widehat{\mathbb{P}}_{2}\), where \(P\left(t, T_{1}\right)\) and \(P\left(t, T_{2}\right)\) are bond prices with maturities \(T_{1}\) and \(T_{2}\).

b) State the expression of the forward rate \(f\left(t, T_{1}, T_{2}\right)\).

c) Compute the dynamics of \(f\left(t, T_{1}, T_{2}\right)\) under the forward measure \(\widehat{\mathbb{P}}_{2}\) with

\[\frac{\mathrm{d} \widehat{\mathbb{P}}_{2}}{\mathrm{~d} \mathbb{P}^{*}}=\frac{1}{P\left(0, T_{2}\right)} \mathrm{e}^{-\int_{0}^{T_{2}} r_{s} d s}\]

d) Compute the price

\[\left(T_{2}-T_{1}\right) \mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T_{2}} r_{s} d s}\left(f\left(T_{1}, T_{1}, T_{2}\right)-\kappa\right)^{+} \mid \mathcal{F}_{t}\right]\]

of an interest rate cap at time \(t \in\left[0, T_{1}\right]\), using the expectation under the forward measure \(\widehat{\mathbb{P}}_{2}\).

e) Compute the dynamics of the swap rate process

\[S\left(t, T_{1}, T_{2}\right)=\frac{P\left(t, T_{1}\right)-P\left(t, T_{2}\right)}{\left(T_{2}-T_{1}\right) P\left(t, T_{2}\right)}, \quad t \in\left[0, T_{1}\right]\]

under \(\widehat{\mathbb{P}}_{2}\).

f) Using (19.33), compute the swaption price \[\left(T_{2}-T_{1}\right) \mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T_{1}} r_{s} d s} P\left(T_{1}, T_{2}\right)\left(S\left(T_{1}, T_{1}, T_{2}\right)-\kappa\right)^{+} \mid \mathcal{F}_{t}\right]\]
on the swap rate \(S\left(T_{1}, T_{1}, T_{2}\right)\) using the expectation under the forward swap measure \(\widehat{\mathbb{P}}_{1,2}\).