An Efficient Physics-Informed Neural Network Solution to the Time-Space Fractional Black-Scholes Equation

This study develops a rigorous analytical and computational framework for solving the time-space-fractional Black–Scholes equation (ts-fBSE), a generalization of the classical Black–Scholes model that captures nonlocal temporal memory and spatial anomalous diffusion in financial markets. Starting from fractional stochastic dynamics driven by Gaussian white noise, we derive the ts-fBSE using generalized Itô–Lévy calculus and establish its well-posedness under appropriate initial and boundary conditions. We demonstrate that the conventional transformation $$y = \ln S + a$$ y = ln S + a does not, in general, reduce the spatial operator to integer order and provide an alternative transformation that yields a constant-coefficient time-fractional BSPDE. The equation is solved using a physics-informed neural network (PINN) incorporating the Grünwald–Letnikov fractional derivative through a stable matrix formulation, eliminating mesh discretization and stability constraints typical of finite-difference methods. The PINN loss functional enforces the operator residual in $$L^2(\Omega )$$ L 2 ( Ω ) augmented by boundary and terminal penalties, trained with a piecewise-constant decay learning rate and a stopping tolerance of $$10^{-4}$$ 10 - 4 . Numerical experiments for European put options validate the accuracy and stability of the method, showing decreasing mean absolute error as the fractional order $$\alpha \rightarrow 1$$ α → 1 . The results confirm that the proposed PINN framework provides a mathematically consistent and computationally robust alternative for solving fractional–stochastic PDEs in quantitative finance, complementing recent developments such as fPINNs and XPINNs.