A Mellin Transform Approach to Black-Scholes Equation for Generalized ML-Payoff Function

This paper develops a novel analytical framework for pricing European put options using a generalized modified-log (ML) payoff function within the Black--Scholes model. By applying the Mellin transform, we derive a closed-form pricing formula that unifies several well-known special cases, including standard log and power-log options. The approach provides mathematical flexibility to represent a wide class of nonlinear payoffs through a finite series expansion $ g(S) = \sum_{i=1}^{n} b_i S^{p_i} $. The study further analyses the sensitivity measures (Greeks) and presents comprehensive graphical interpretations illustrating their classical behaviour under the generalized model. Numerical experiments demonstrate strong agreement between the Mellin-based pricing formula and the conventional Black--Scholes model in limiting cases, while achieving enhanced accuracy for complex payoffs. The results confirm the robustness, convergence, and computational stability of the proposed formulation, offering a unified and efficient tool for pricing exotic, path-independent derivatives in emerging financial markets. AMS Mathematics Subject Classification: 91B25, 91G20