Consider two bonds with maturities \(T\) and \(S\), with prices \(P(t, T)\) and \(P(t, S)\) given by

\[\frac{d P(t, T)}{P(t, T)}=r_{t} d t+\zeta_{t}^{T} d W_{t}\]

and

\[\frac{d P(t, S)}{P(t, S)}=r_{t} d t+\zeta_{t}^{S} d W_{t}\]

where \(\left(\zeta^{T}(s)\right)_{s \in[0, T]}\) and \(\left(\zeta^{S}(s)\right)_{s \in[0, S]}\) are deterministic volatility functions of time.

a) Show, using Itô's formula, that

\[d\left(\frac{P(t, S)}{P(t, T)}\right)=\frac{P(t, S)}{P(t, T)}\left(\zeta^{S}(t)-\zeta^{T}(t)\right) d \widehat{W}_{t}\]

where \(\left(\widehat{W}_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under \(\widehat{\mathbb{P}}\).

b) Show that

\[P(T, S)=\frac{P(t, S)}{P(t, T)} \exp \left(\int_{t}^{T}\left(\zeta^{S}(s)-\zeta^{T}(s)\right) d \widehat{W}_{s}-\frac{1}{2} \int_{t}^{T}\left|\zeta^{S}(s)-\zeta^{T}(s)\right|^{2} d s\right) .\]

Let \(\widehat{\mathbb{P}}\) denote the forward measure associated to the numéraire

\[N_{t}:=P(t, T), \quad 0 \leqslant t \leqslant T\]

c) Show that for all \(S, T>0\) the price at time \(t\)
\[\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s}(P(T, S)-\kappa)^{+} \mid \mathcal{F}_{t}\right]\]
of a bond call option on \(P(T, S)\) with payoff \((P(T, S)-\kappa)^{+}\)is equal to \[\begin{align*}
\mathbb{E}^{*} & {\left[\mathrm{e}^{-\int_{t}^{T} r_{s} d s}(P(T, S)-\kappa)^{+} \mid \mathcal{F}_{t}\right] } \tag{16.37}\\
& =P(t, S) \Phi\left(\frac{v}{2}+\frac{1}{v} \log \frac{P(t, S)}{\kappa P(t, T)}\right)-\kappa P(t, T) \Phi\left(-\frac{v}{2}+\frac{1}{v} \log \frac{P(t, S)}{\kappa P(t, T)}\right)
\end{align*}\]
where \[v^{2}=\int_{t}^{T}\left|\zeta^{S}(s)-\zeta^{T}(s)\right|^{2} d s\]

d) Compute the self-financing hedging strategy that hedges the bond option using a portfolio based on the assets \(P(t, T)\) and \(P(t, S)\).