A mathematical perspective on the evolution of option pricing models and digital engineering for finance

Abstract

This article presents a comprehensive study of conformable calculus and its application to financial modeling, with particular emphasis on the Black–Scholes framework for pricing European options. Conformable derivatives extend classical differentiation while preserving fundamental properties such as linearity and the chain rule, making them suitable for analytical and numerical analysis. By introducing conformable time dynamics into stochastic processes, a generalized Black–Scholes partial differential equation is derived that accounts for nonstandard temporal behavior in financial markets. An explicit analytical pricing formula for European options is obtained and shown to reduce to the classical Black–Scholes formula as a limiting case. The influence of the conformable order on option prices is examined, revealing its effect on volatility sensitivity and time-to-maturity characteristics. To support the theoretical developments, a numerical example is presented that illustrates the impact of conformable parameters on option valuation. The findings indicate that conformable calculus offers a flexible and effective framework for enhancing traditional option pricing models and capturing complex market dynamics.