Consider an asset price $S_{t}$ given by $S_{t}=$ $S_{0} \mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}, t \geqslant 0$, where $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$is a standard Brownian motion, with $r \geqslant 0$ and $\sigma>0$.

a) Compute the average $\mathbb{E}^{*}\left[m_{0}^{T}ight]$ of the minimum $m_{0}^{T}:=\min _{t \in[0, T]} S_{t}$ of $\left(S_{t}ight)_{t \in[0, T]}$ over $[0, T]$.

b) Compute the expected payoff $\mathbb{E}\left[\left(K-\min _{t \in[0, T]} S_{t}ight)^{+}ight]$for $r>0$. Using a finite expiration American put option pricer, compare the American put option price to the above expected payoff.

c) Compute the expected payoff $\mathbb{E}\left[\left(K-\min _{t \in[0, T]} S_{t}ight)^{+}ight]$for $r=0$.