Distributional Mellin calculus in $\mathbb{C}^n$, with applications to option pricing

TL;DR

This paper explores advanced Mellin transform techniques in complex spaces and applies them to various option pricing models, including European and American options, providing new mathematical tools for financial modeling.

Contribution

It introduces distributional Mellin calculus in multiple complex variables and demonstrates its application to complex option pricing problems.

Findings

Effective evaluation of European option models using Mellin techniques

Novel methods for American option exercise price analysis

Connections established between Mellin transforms and Laplace integral evaluations

Abstract

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.