Numerical analysis of American option pricing in a two-asset jump-diffusion model

Paper: arXiv 2410.04745

Abstract

This paper addresses an important gap in rigorous numerical treatments for pricing American options under correlated two-asset jump-diffusion models using the viscosity solution framework, with a particular focus on the Merton model. The pricing of these options is governed by complex two-dimensional (2-D) variational inequalities that incorporate cross-derivative terms and nonlocal integro-differential terms due to the presence of jumps. Existing numerical methods, primarily based on finite differences, often struggle with preserving monotonicity in the approximation of cross-derivatives, a key requirement for ensuring convergence to the viscosity solution. In addition, these methods face challenges in accurately discretizing 2-D jump integrals. We introduce a novel approach to effectively tackle the aforementioned variational inequalities while seamlessly handling cross-derivative terms and nonlocal integro-differential terms through an efficient and straightforward-to-implement monotone integration scheme. Within each timestep, our approach explicitly enforces the inequality constraint, resulting in a 2-D Partial Integro-Differential Equation (PIDE) to solve. Its solution is expressed as a 2-D convolution integral involving the Green’s function of the PIDE. We derive an infinite series representation of this Green’s function, where each term is non-negative and computable. This facilitates the numerical approximation of the PIDE solution through a monotone integration method. To enhance efficiency, we develop an implementation of this monotone scheme via FFTs, exploiting the Toeplitz matrix structure. The proposed method is proved to be both $\ell_{"\infty"} $-stable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of the variational inequality. Extensive numerical results validate the effectiveness and robustness of our approach.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical concepts including viscosity solutions, infinite series representations of Green’s functions, and rigorous stability proofs, indicating high mathematical complexity. While it lacks actual code or backtest results, the detailed numerical scheme design and validation for a specific, computationally intensive problem (American options under 2-D jump-diffusion) suggests strong empirical rigor focused on method implementation rather than pure data analysis.

Research Flowchart

  flowchart TD
  A["Research Goal: Robust Numerical Pricing<br>American Options in 2-Asset Jump-Diffusion"] --> B["Formulate 2-D Variational Inequality<br>with Nonlocal Jumps & Cross-Derivatives"]
  B --> C["Novel Monotone Integration Scheme<br>Decompose VI into Constraint & 2-D PIDE"]
  C --> D{"Solve 2-D PIDE via<br>Convolution with Green's Function"}
  D --> E["Compute Green's Function<br>Infinite Series Representation"]
  E --> F["Numerical Approximation<br>Fast Fourier Transform FFT Optimization"]
  F --> G["Validation & Results<br>Monotonicity, Stability & Convergence"]
  G --> H["Outcome: Efficient & Robust<br>Algorithm for American Option Pricing"]
  
  style A fill:#e1f5fe,stroke:#01579b
  style H fill:#e8f5e9,stroke:#2e7d32
  style C fill:#fff3e0,stroke:#ef6c00
  style F fill:#f3e5f5,stroke:#4a148c

Cited References (533)

1. An efficient numerical method for American options and their Greeks under the two-asset Kou jump-diffusion model (2024) — link
2. Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data (2010)
3. High-order WENO schemes for Hamilton–Jacobi equations on triangular meshes (2003)
4. L' e vy Processes: Large Deviations (2005)
5. A Pointwise Convergent Numerical Integration Method for Guaranteed Lifelong Withdrawal Benefits under Stochastic Volatility (2023) — link
6. An $\epsilon$-Monotone Fourier Method for Guaranteed Minimum Withdrawal Benefit (GMWB) as a Continuous Impulse Control Problem (2022) — link
7. Fourier Neural Network Approximation of Transition Densities in Finance — → 2309.03966
8. A survey on M-matrices (1974)

4 of 533 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.