A Physics informed neural network approach for solving time fractional Black-Scholes partial differential equations

Abstract

We present a novel approach for solving time fractional Black-Scholes partial differential equations (tfBSPDEs) using Physics Informed Neural Network (PINN) approach. Traditional numerical methods are faced with challenges in solving fractional PDEs due to the non-locality and non-differentiability nature of fractional derivative operators. By leveraging the ideas of Riemann sums and the refinement of tagged partitions of the time domain, we show that fractional derivatives can directly be incorporated into the loss function when applying the PINN approach to solving tfBSPDEs. The approach allows for the simultaneous learning of the underlying process dynamics and the involved fractional derivative operator without a need for the use of numerical discretization of the fractional derivatives. Through some numerical experiments, we demonstrate that, the PINN approach is efficient, accurate and computationally inexpensive particularly when dealing with high frequency and noisy data. This work augments the understanding between advanced mathematical modeling and machine learning techniques, contributing to the body of knowlege on the advancement of accurate derivative pricing models.