Assume that the price \(P(t, T)\) of a zero-coupon bond with maturity \(T>0\) is modeled as

\[P(t, T)=\mathrm{e}^{-\mu(T-t)+X_{t}^{T}}, \quad t \in[0, T],\]

where \(\left(X_{t}^{T}\right)_{t \in[0, T)}\) is the solution of the stochastic differential equation

\[d X_{t}^{T}=\sigma d B_{t}-\frac{X_{t}^{T}}{T-t} d t, \quad t \in[0, T)\]

under the initial condition \(X_{0}^{T}=0\), i.e.

\[X_{t}^{T}=(T-t) \int_{0}^{t} \frac{\sigma}{T-s} d B_{s}, \quad 0 \leqslant t

with \(\mu, \sigma>0\).

a) Show that the terminal condition \(P(T, T)=1\) is satisfied.

b) Compute the forward rate

\[f(t, T, S)=-\frac{1}{S-T}(\log P(t, S)-\log P(t, T))\]

c) Compute the instantaneous forward rate

\[f(t, T)=-\lim _{S \searrow T} \frac{1}{S-T}(\log P(t, S)-\log P(t, T)) .\]

d) Show that the limit \(\lim _{T \searrow t} f(t, T)\) does not exist in \(L^{2}(\Omega)\).

e) Show that \(P(t, T)\) satisfies the stochastic differential equation \[\frac{d P(t, T)}{P(t, T)}=\sigma d B_{t}+\frac{\sigma^{2}}{2} d t-\frac{\log P(t, T)}{T-t} d t, \quad t \in[0, T] .\]

f) Rewrite the equation of Question (e) as \[\frac{d P(t, T)}{P(t, T)}=\sigma d B_{t}+r_{t}^{T} d t, \quad t \in[0, T]\]
where \(\left(r_{t}^{\mathrm{T}}\right)_{t \in[0, T]}\) is a process to be determined.
g) Show that we have the expression \[P(t, T)=\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s}^{\mathrm{T}} d s} \mid \mathcal{F}_{t}\right], \quad 0 \leqslant t \leqslant T\]
h) Compute the conditional Radon-Nikodym density \[\mathbb{E}^{*}\left[\left.\frac{\mathrm{d} \mathbb{P}_{T}}{\mathrm{~d} \mathbb{P}^{*}} \rightvert\, \mathcal{F}_{t}\right]=\frac{P(t, T)}{P(0, T)} \mathrm{e}^{-\int_{0}^{t} r_{s}^{T} d s}\]
of the forward measure \(\mathbb{P}_{T}\) with respect to \(\mathbb{P}^{*}\).
i) Show that the process \[\widehat{B}_{t}:=B_{t}-\sigma t, \quad 0 \leqslant t \leqslant T\]
is a standard Brownian motion under \(\mathbb{P}_{T}\).
j) Compute the dynamics of \(X_{t}^{S}\) and \(P(t, S)\) under \(\mathbb{P}_{T}\).
Show that \[-\mu(S-T)+\sigma(S-T) \int_{0}^{t} \frac{1}{S-s} d B_{s}=\frac{S-T}{S-t} \log P(t, S) .\]
k) Compute the bond option price \[\mathbb{E}^{*}\left[\mathrm{e}^{-\int_{t}^{T} r_{s}^{\mathrm{T}} d s}(P(T, S)-K)^{+} \mid \mathcal{F}_{t}\right]=P(t, T) \mathbb{E}_{T}\left[(P(T, S)-K)^{+} \mid \mathcal{F}_{t}\right]\]
\(0 \leqslant tGiven \(X\) a Gaussian random variable with mean \(m\) and variance \(v^{2}\) given \(\mathcal{F}_{t}\), we have:
\[\begin{aligned}
\mathbb{E}\left[\left(\mathrm{e}^{X}-\kappa\right)^{+} \mid \mathcal{F}_{t}\right]= & \mathrm{e}^{m+v^{2} / 2} \Phi\left(\frac{1}{v}\left(m+v^{2}-\log \kappa\right)\right) \\
& -\kappa \Phi\left(\frac{1}{v}(m-\log \kappa)\right)
\end{aligned}\]