Exercise 7.20 Let \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)be a standard Brownian motion generating a filtration \(\left(\mathcal{F}_{t}ight)_{t \in \mathbb{R}_{+}}\). Recall that for \(f \in \mathcal{C}^{2}\left(\mathbb{R}_{+} \times \mathbb{R}ight)\), Itô's formula for \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)reads

\[
\begin{aligned}
f\left(t, B_{t}ight)= & f\left(0, B_{0}ight)+\int_{0}^{t} \frac{\partial f}{\partial s}\left(s, B_{s}ight) d s \\
& +\int_{0}^{t} \frac{\partial f}{\partial x}\left(s, B_{s}ight) d B_{s}+\frac{1}{2} \int_{0}^{t} \frac{\partial^{2} f}{\partial x^{2}}\left(s, B_{s}ight) d s
\end{aligned}
\]

a) Let \(r \in \mathbb{R}, \sigma>0, f(x, t)=\mathrm{e}^{r t+\sigma x-\sigma^{2} t / 2}\), and \(S_{t}=f\left(t, B_{t}ight)\). Compute \(d f\left(t, B_{t}ight)\) by Itô's formula, and show that \(S_{t}\) solves the stochastic differential equation

\[
d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t},
\]

where \(r>0\) and \(\sigma>0\).

b) Show that

\[
\mathbb{E}\left[\mathrm{e}^{\sigma B_{T}} \mid \mathcal{F}_{t}ight]=\mathrm{e}^{\sigma B_{t}+(T-t) \sigma^{2} / 2}, \quad 0 \leqslant t \leqslant T
\]

Hint: Use the independence of increments of \(\left(B_{t}ight)_{t \in[0, T]}\) in the time splitting decomposition

\[
B_{T}=\left(B_{t}-B_{0}ight)+\left(B_{T}-B_{t}ight),
\]

and the Gaussian moment generating function \(\mathbb{E}\left[\mathrm{e}^{\alpha X}ight]=\mathrm{e}^{\alpha^{2} \eta^{2} / 2}\) when \(X \simeq \mathcal{N}\left(0, \eta^{2}ight)\).

c) Show that the process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)satisfies

\[
\mathbb{E}\left[S_{T} \mid \mathcal{F}_{t}ight]=\mathrm{e}^{(T-t) r} S_{t}, \quad 0 \leqslant t \leqslant T
\]

d) Let \(C=S_{T}-K\) denote the payoff of a forward contract with exercise price \(K\) and maturity \(T\). Compute the discounted expected payoff

\[
V_{t}:=\mathrm{e}^{-(T-t) r} \mathbb{E}\left[C \mid \mathcal{F}_{t}ight]
\]

e) Find a self-financing portfolio strategy \(\left(\xi_{t}, \eta_{t}ight)_{t \in \mathbb{R}_{+}}\)such that

\[
V_{t}=\xi_{t} S_{t}+\eta_{t} A_{t}, \quad 0 \leqslant t \leqslant T
\]

where \(A_{t}=A_{0} \mathrm{e}^{r t}\) is the price of a riskless asset with fixed interest rate \(r>0\). Show that it recovers the result of Exercise 6.7-(c).

f) Show that the portfolio allocation \(\left(\xi_{t}, \eta_{t}ight)_{t \in[0, T]}\) found in Question (e) hedges the payoff \(C=S_{T}-K\) at time \(T\), i.e. show that \(V_{T}=C\).