Jamshidian's trick (Jamshidian (1989)). Consider a family (left(Pleft(t, T_{l} ight) ight)_{l=i, ldots, j}) of bond prices defined

Jamshidian's trick (Jamshidian (1989)). Consider a family \(\left(P\left(t, T_{l}\right)\right)_{l=i, \ldots, j}\) of bond prices defined from a short rate process \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\). We assume that the bond prices are functions \(P\left(T_{i}, T_{l+1}\right)=F_{l+1}\left(T_{i}, r_{T_{i}}\right)\) of \(r_{T_{i}}\) that are increasing in the variable \(r_{T_{i}}\), for all \(l=i, i+1, \ldots, j-1\).

a) Compute the price \(P\left(t, T_{i}, T_{j}\right)\) of the annuity numéraire paying coupons \(c_{i+1}, \ldots, c_{j}\) at times \(T_{i+1}, \ldots, T_{j}\) in terms of the bond prices

\[P\left(t, T_{i+1}\right), \ldots, P\left(t, T_{j}\right)\]

b) Show that the payoff

\[\left(P\left(T_{i}, T_{i}\right)-P\left(T_{i}, T_{j}\right)-\kappa P\left(T_{i}, T_{i}, T_{j}\right)\right)^{+}\]

of a European swaption can be rewrittten as

\[\left(1-\kappa \sum_{l=i}^{j-1} \tilde{c}_{l+1} P\left(T_{i}, T_{l+1}\right)\right)^{+}\]

by writing \(\tilde{c}_{l}\) in terms of \(c_{l}, l=i+1, \ldots, j\).

c) Assuming that the bond prices are functions \(P\left(T_{i}, T_{l+1}\right)=F_{l}\left(T_{i}, r_{T_{i}}\right)\) of \(r_{T_{i}}\) that are increasing in the variable \(r_{T_{i}}\), for all \(l=i, \ldots, j-1\), show, choosing \(\gamma_{\kappa}\) such that \[\kappa \sum_{l=i}^{j-1} c_{l+1} F_{l+1}\left(T_{i}, \gamma_{\kappa}\right)=1\]
that the European swaption with payoff \[\left(P\left(T_{i}, T_{i}\right)-P\left(T_{i}, T_{j}\right)-\kappa P\left(T_{i}, T_{i}, T_{j}\right)\right)^{+}=\left(1-\kappa \sum_{l=i}^{j-1} c_{l+1} P\left(T_{i}, T_{l+1}\right)\right)^{+},\]
where \(c_{j}\) contains the final coupon payment, can be priced as a weighted sum of bond put options under the forward measure \(\widehat{\mathbb{P}}_{i}\) with numéraire \(N_{t}^{(i)}:=P\left(t, T_{i}\right)\).