  We discuss several aspects of Mellin transform, including distributional
Mellin transform and inversion of multiple Mellin-Barnes integrals in
$\mathbb{C}^n$ and its connection to residue expansion or evaluation of Laplace
integrals. These mathematical concepts are demonstrated on several
option-pricing models. This includes European option models such as
Black-Scholes or fractional-diffusion models, as well as evaluation of
quantities related to the optimal exercise price of American options.
