Using From Relation (10.11) in Proposition 10.3 and the Jacobian change of variable formula, assuming $S_{0}>0$, compute the joint probability density function of geometric Brownian motion $S_{T}:=S_{0} \mathrm{e}^{\sigma W_{T}+\left(r-\sigma^{2} / 2ight) T}$ and its maximum

$$
M_{0}^{T}:=\operatorname{Max}_{t \in[0, T]} S_{t}=S_{0} \operatorname{Max}_{t \in[0, T]} \mathrm{e}^{\sigma W_{t}+\left(r-\sigma^{2} / 2ight) t}
$$