This study aims to investigate the European Call Option in a fractional way and the reaction of the Hamilton-Jacobi differential equation with the European Call Option. We explore the applications of Mellin transforms as a method for solving such fractional differential equations. We can address initial value problems defined for arbitrary orders of differentiation and integration in fractional calculus that allows us to understand the solution's evolution, encompassing the values established in classical analysis. The theory of options permits the connection of physics and finance, particularly through dynamic systems and their relationship with the Hamilton-Jacobi equation. The European Call Option model exhibits energy in its Hamiltonian, leading to investigations of the Lagrangian and its potential functions within the European Call Option model. Caputo’s and Riemann-Liouville's fractional derivatives are crucial for differential equations with integer and non-integer initial conditions, respectively. This new perspective on calculus challenges traditional analysis, explores new horizons, and demands a redefinition of classical approaches. Understanding the efficacy of integro-differential operators is pivotal for advancing research, particularly in interpreting financial phenomena using classical calculus. The Mittag-Leffler function proves valuable for algebraic purposes and inverse transformations such as Mellin. Keywords: European call option, the Hamilton-Jacobi equation, the Mittag-Leffler function, the Mellin transform, the Riemann-Liouville fractional derivative. https://doi.org/10.55463/issn.1674-2974.50.7.13