A Novel Fractional Stochastic Model Equipped With ψ-Caputo Fractional Derivative in a Financial Market

Abstract

The present article describes an improved version of the Black–Scholes model, an important model in finance used for option pricing. To overcome the shortcomings of this traditional model caused by the assumptions and simplification of the original model itself, we use the fractional (Formula presented.) -Caputo derivative to describe the real movement of the market more accurately. The (Formula presented.) function adds extra freedom to our model so that the resulting model can better describe the market price. We prove the existence and specificity of the solution for the SDE linked to this model. Further, we propose a numerical scheme using the modified Euler's method to solve the SDE and to consider the fractional derivative. This research provides a stronger basis for the specified financial modeling, especially for the systems that may have some dynamics the basic methods cannot capture.