Finite Element Method for HJB in Option Pricing with Stock Borrowing Fees

ArXiv ID: 2501.02327 “View on arXiv”

Authors: Unknown

Abstract

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 study, we apply a finite element approach with a non-uniform mesh for the task of option pricing with stock borrowing fees, leading to an HJB equation that bypasses analytical optimal control in favor of direct PDE discretization. The time integration employs the theta-scheme, with initial modifications following Rannacher`s procedure. A Newton-type algorithm is applied to address the penalty-like term at each time step. Numerical experiments are conducted, demonstrating consistency with a benchmark problem and showing a strong match. The CPU time needed to reach the desired results favors P2-FEM over FDM and linear P1-FEM, with P2-FEM displaying superior convergence. This paper 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.

Keywords: HJB equation, finite element method (FEM), optimal control, derivatives pricing, numerical schemes, derivatives

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats
  • Why: The paper involves advanced mathematics, including nonlinear HJB PDEs, Sobolev spaces, weak formulations, and high-order finite element methods (P2-FEM), placing it in the high math complexity range. However, the empirical section appears to rely on a benchmark comparison and CPU time metrics rather than extensive real-world backtesting or large-scale dataset validation, indicating lower empirical rigor.
  flowchart TD
    A["Research Goal<br>Price options with frictions via HJB<br>using FEM"] --> B["Model Formulation<br>HJB PDE with penalty term for borrowing fees"]
    B --> C["Discretization Strategy<br>Non-uniform mesh FEM<br>Time: Theta-scheme w/ Rannacher init"]
    C --> D["Solver<br>Newton-type algorithm per time step"]
    D --> E["Computation<br>Compare P2-FEM vs. FDM vs. P1-FEM"]
    E --> F["Findings<br>Consistent with benchmark<br>P2-FEM shows superior convergence & efficiency"]