Let \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)denote a standard Brownian motion.

a) Compute the expected value

\[
\mathbb{E}\left[\operatorname{Max}_{t \in[0,1]} S_{t}ight]=\mathbb{E}\left[\mathrm{e}^{\sigma \operatorname{Max}_{t \in[0,1]}\left(B_{t}-\sigma t / 2ight)}ight]
\]

b) Compute the "optimal exercise" price \(\mathbb{E}\left[\left(S_{0} \operatorname{Max}_{t \in[0,1]} \mathrm{e}^{\sigma B_{t}-\sigma^{2} t / 2}-Kight)^{+}ight]\)of a finite expiration American call option with \(S_{0} \geqslant K\).