<p>In mathematical finance, many derivatives from markets with frictions can be formulated as optimal control problems in the HJB framework. Analytical optimal control can result in highly nonlinear PDEs, which might yield unstable numerical results. Accurate and convergent numerical schemes are essential to leverage the benefits of the hedging process. In this paper, we apply finite element approach with non-uniform mesh for option pricing with stock borrowing fees, leading to an HJB equation that enables direct numerical treatment, circumventing the need for closed-form optimal controls. The time integration employs the theta-scheme with Rannacher’s smoothing procedure for initial time steps. A Newton-type algorithm efficiently handles the penalty-like term (which arises from the optimal control of stock borrowing fees and introduces non-differentiable nonlinearity) at each time step. Numerical experiments demonstrate consistency with benchmark problems. The proposed framework achieves superior computational efficiency, with P2-FEM outperforming both FDM and linear P1-FEM in CPU time while displaying enhanced convergence. This work presents an efficient alternative framework for the HJB problem and contributes to the literature by introducing a finite element method (FEM)-based solution for HJB applications in mathematical finance.</p>