a) Consider the solution \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)of the stochastic differential equation

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

For which value \(\alpha_{M}\) of \(\alpha\) is the discounted price process \(\widetilde{S}_{t}=\mathrm{e}^{-r t} S_{t}, 0 \leqslant t \leqslant T\), a martingale under \(\mathbb{P}\) ?

b) For each value of \(\alpha\), build a probability measure \(\mathbb{P}_{\alpha}\) under which the discounted price process \(\widetilde{S}_{t}=\mathrm{e}^{-r t} S_{t}, 0 \leqslant t \leqslant T\), is a martingale.

c) Compute the arbitrage-free price

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

at time \(t \in[0, T]\) of the contingent claim with payoff \(\exp \left(S_{T}ight)\), and recover the result of Exercise 6.11.

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

e) Check that this strategy is self-financing.