a) Bachelier (1900) model. Solve the stochastic differential equation

\[
\begin{equation*}
d S_{t}=\alpha S_{t} d t+\sigma d B_{t} \tag{6.41}
\end{equation*}
\]
in terms of \(\alpha, \sigma \in \mathbb{R}\), and the initial condition \(S_{0}\).

b) Write down the Bachelier PDE satisfied by the function \(C(t, x)\), where \(C\left(t, S_{t}ight)\) is the price at time \(t \in[0, T]\) of the contingent claim with claim payoff \(C=\phi\left(S_{T}ight)=\exp \left(S_{T}ight)\), and identify the process Delta \(\left(\xi_{t}ight)_{t \in[0, T]}\) that hedges this claim.

c) Solve the Black-Scholes PDE of Question

(b) with the terminal condition \(\phi(x)=\mathrm{e}^{x}\), \(x \in \mathbb{R}\).
Search for a solution of the form \[
\begin{equation*}
C(t, x)=\exp \left(-(T-t) r+x h(t)+\frac{\sigma^{2}}{4 r}\left(h^{2}(t)-1ight)ight) \tag{6.42}
\end{equation*}
\]
where \(h(t)\) is a function to be determined, with \(h(T)=1\).

d) Compute the portfolio strategy \(\left(\xi_{t}, \eta_{t}ight)_{t \in[0, T]}\) that hedges the contingent claim with claim payoff \(C=\exp \left(S_{T}ight)\).