Mellin transform as a distribution framework for option pricing

Abstract

This work establishes Mellin Transform as a space of distributions for option pricing. The Mellin Transform was not just treated as a computational tool but also as a distributional representative of asset prices. Unlike classical approaches such as the Black–Scholes model, which rely on smooth payoff structures and Gaussian assumptions, Option payoffs were treated as generalized functions and Mellin-based pricing formula was derived. A second-order asymptotic expansion is introduced via the saddle-point method, capturing higher order effects such as skewness and kurtosis. Empirical observed daily properties of financial returns are captured in a concrete setting.