New Illustrative Applications of Integral Transforms to Financial Models with Different Fractional Derivatives

Abstract We investigate a couple of different financial/economic models based on market equilibrium and option pricing with three different fractional derivatives in this paper. We obtain the fundamental solutions of the models by Sumudu transform and Laplace transform. We demonstrate our results by illustrative figures to point out the difference between the fractional operators that have power kernel, exponential kernel and Mittag-Leffler kernel. We prove the efficiency and accuracy of the Sumudu transform and decomposition series method constructed by the Laplace transform in providing the solutions of several different linear/nonlinear financial models by considering the theoretical results and illustrative applications. It seems that the proposed method is an efficient way to solve such problems that contain different types of fractional operators and one is able to point out the differences between these mentioned operators. One of the valuable features of the method is the possibility of using it in solving other similar equations including fractional derivatives having a singular or nonsigular kernel. This paper also suggests a good initiative and profitable tool for those who want to invest in these types of options either individually or institutionally.