On a Stable Method for Option Pricing: Discontinuous Petrov-Galerkin Method for Option Pricing and Sensitivity Analysis

TL;DR

This paper introduces a stable Discontinuous Petrov-Galerkin (DPG) method for option pricing and sensitivity analysis within the Black-Scholes model, demonstrating its effectiveness through numerical experiments and providing an open-source HPC implementation.

Contribution

The paper adapts the DPG methodology for various types of options, deriving primal and ultra-weak formulations, and offers a publicly available high-performance C++ code.

Findings

Demonstrates convergence, stability, and efficiency of the DPG method for different options.

Provides a high-performance C++ implementation for practical use.

Validates the method through extensive numerical experiments.

Abstract

The discontinuous Petrov Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan introduced in their first paper has been widely used for problems in computational mechanics. In this investigation, we propose the DPG method for option pricing and sensitivity analysis under the basic Black-Scholes model. In this investigation, primal and ultra-weak formulation of the DPG method is derived for Vanilla options, American options, Asian options, and Barrier options. A wide range of standard numerical experiments is conducted to examine the convergence, stability, and efficiency of the proposed method for each one of the options separately. Besides, a C++ high performance (HPC) code for option pricing with the DPG method is developed which is available to the public to customize it for option pricing problems or other related problems.