<p>In this study, we present a computational approach to numerically solve the Black–Scholes partial differential equations (PDEs) related to European option pricing and its extensions. We have addressed scenarios where traditional methods have failed to yield satisfactory results due to the specific characteristics of market parameters. To tackle this challenge, we employed the streamline-diffusion finite element method (SDFEM). We discretized the time domain using a uniform mesh, applying both the Crank–Nicolson scheme and the backward-Euler scheme separately to approximate the temporal component. For the spatial variable discretization, we utilized the SDFEM on various nonuniform meshes. We establish theoretical results ensuring the stability and convergence of the proposed method. Numerical experiments are included to validate the theoretical convergence results and to compare our proposed technique with traditional numerical methods as well as existing literature. Furthermore, we demonstrate the additional application of our proposed technique by examining the lookback option.</p>