a) Give the probability density function of the maximum of drifted Brownian motion \(\operatorname{Max}_{t \in[0,1]}\left(B_{t}+\sigma t / 2ight)\).

b) Taking \(S_{t}:=\mathrm{e}^{\sigma B_{t}-\sigma^{2} t / 2}\), compute the expected value

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

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