Analytical Solution of the Time–Fractional Black–Scholes Equation Using the Caputo–Fabrizio Derivative via the Sumudu Transform

Abstract

We derive an analytical representation for the solution of the Time–Fractional Black–Scholes Equation in which the usual first-order time derivative is replaced by the Caputo–Fabrizio (CF) fractional derivative. The Sumudu transform is used to convert the Time–Fractional PDE into a family of second-order ordinary differential equations in the asset variable. We give a full derivation of the Sumudu transform of the CF operator, solve the transformed ODE by reducing it to constant-coefficient form in the logarithmic asset variable, construct the corresponding Green’s function (resolvent), and present the transform-domain solution representation. Numerical inversion of the Sumudu transform is described and implemented to compute option prices. Numerical results are presented for two benchmark examples and one additional test case for fractional orders α = 0.25, 0.5, 0.75, 1.0, using several spatial grid sizes. The results demonstrate the convergence, accuracy, and stability of the proposed method.