Given the price process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)defined as the geometric Brownian motion

\[
S_{t}:=S_{0} \mathrm{e}^{\sigma B_{t}+\left(r-\sigma^{2} / 2ight) t}, \quad t \geqslant 0
\]

price the option with payoff function \(\phi\left(S_{T}ight)\) by writing \(\mathrm{e}^{-r T} \mathbb{E}^{*}\left[\phi\left(S_{T}ight)ight]\) as an integral with respect to the lognormal probability density function.