Approximate Analytical Solution of the Black–Scholes Model with Two Assets Based on the ABC Time-Fractional Derivative

The classical Black–Scholes model assumes Markovian dynamics and cannot capture the long-range dependence and gradual memory decay observed in real markets. We formulate the two-dimensional time-fractional Black–Scholes equation for a European put on a weighted basket of two correlated assets under the Atangana–Baleanu–Caputo (ABC) derivative, whose non-singular Mittag-Leffler kernel models distributed, fading memory more faithfully than the singular Riemann–Liouville and Caputo kernels and the localized Caputo–Fabrizio kernel. A closed-form approximate analytical solution is derived via the Laplace homotopy perturbation method. We prove a convergence theorem with an explicit geometric error bound, and show that the series solves the associated Atangana–Baleanu integral equation exactly and the differential equation up to an explicit, decaying initial-layer term that vanishes as ξ→1. We further prove that, for the basket payoff, the closed-form price is independent of the inter-asset correlation. The solution reduces to the classical two-asset price deep in the money as ξ→1, agreeing with a Monte Carlo benchmark to within 0.1% in that regime, where the approximation is valid. The contribution combines three elements: the two-asset setting, the non-singular Mittag-Leffler kernel, and a closed-form solution.