Reliable Numerical Pricing of Multi–Asset European Options Using a Weak Galerkin FEM for Generalized Black–Scholes Models

ABSTRACT In this study, we present a weak Galerkin finite element method (WGFEM) for numerically solving the two–dimensional Black–Scholes model governing multi‐asset European options. The model's complexity stems from a non‐zero cross‐derivative factor representing asset correlation. The proposed WGFEM framework is tailored to handle this complexity by utilizing discrete weak gradients on triangular meshes, providing flexibility and accuracy. We use implicit time discretization to ensure the fully‐discrete scheme is stable. This method is formulated to accommodate the correlation term without artificial stabilization, maintaining consistency with the continuous model. Numerical experiments on basket option payoffs demonstrate the scheme's effectiveness, confirming optimal convergence rates and robustness. Its real‐world viability is shown through two market‐oriented case studies: pricing a multi‐asset ETF basket option using historical data and validating the fundamental Black–Scholes model against market dynamics. The proposed approach offers a reliable tool for high‐dimensional option pricing, laying the groundwork for future extensions to American and path‐dependent options.