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

TL;DR

This paper introduces a finite element method with a non-uniform mesh for solving HJB equations in option pricing with stock borrowing fees, offering improved convergence and efficiency over traditional methods.

Contribution

It presents a novel FEM-based numerical scheme for HJB equations in finance, bypassing analytical solutions and demonstrating superior convergence and computational performance.

Findings

P2-FEM outperforms FDM and linear P1-FEM in CPU time and convergence.

Numerical experiments show strong agreement with benchmark problems.

The method provides a stable and accurate alternative for HJB PDEs in finance.

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.