A novel numerical scheme for Black-Scholes PDEs modeling pricing securities

This article introduces an efficient numerical method for solving the Black-Scholes partial differential equation (PDE) that governs European options. The methodology employs the backward Euler scheme to discretize the time derivative and incorporates the non-symmetric interior penalty Galerkin method for handling the spatial derivatives. The study aims to determine optimal order error estimates in the $L^2$-norm and discrete energy norm. In addition, the proposed method is used to determine Greeks in option pricing. We validate the theoretical results presented in this work with numerical experiments.