American options valuation in time-dependent jump-diffusion models via integral equations and characteristic functions

ArXiv ID: 2506.18210 “View on arXiv”

Authors: Andrey Itkin

Abstract

Despite significant advancements in machine learning for derivative pricing, the efficient and accurate valuation of American options remains a persistent challenge due to complex exercise boundaries, near-expiry behavior, and intricate contractual features. This paper extends a semi-analytical approach for pricing American options in time-inhomogeneous models, including pure diffusions, jump-diffusions, and Levy processes. Building on prior work, we derive and solve Volterra integral equations of the second kind to determine the exercise boundary explicitly, offering a computationally superior alternative to traditional finite-difference and Monte Carlo methods. We address key open problems: (1) extending the decomposition method, i.e. splitting the American option price into its European counterpart and an early exercise premium, to general jump-diffusion and Levy models; (2) handling cases where closed-form transition densities are unavailable by leveraging characteristic functions via, e.g., the COS method; and (3) generalizing the framework to multidimensional diffusions. Numerical examples demonstrate the method’s efficiency and robustness. Our results underscore the advantages of the integral equation approach for large-scale industrial applications, while resolving some limitations of existing techniques.

Keywords: American options, Volterra integral equations, Lévy processes, option pricing, semi-analytical methods

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper heavily relies on advanced mathematical concepts like Volterra integral equations, characteristic functions, and stochastic calculus, placing it firmly in the high math complexity range. While it claims industrial applicability and efficiency, the excerpt focuses on theoretical derivations and numerical examples without presenting concrete backtesting results, statistical metrics, or code implementations, resulting in low empirical rigor.

  flowchart TD
    A["Research Goal:<br>American option pricing in jump-diffusion/Levy models<br>(Time-inhomogeneous)"] --> B["Key Methodology:<br>Derive Volterra Integral Equation<br>via Decomposition (Premium)"]
    B --> C["Computational Inputs:<br>Transition Density /<br>Characteristic Function"]
    C --> D{"Available Closed Form?"}
    D -- Yes --> E["Numerical Solution<br>Volterra Equation"]
    D -- No --> F["Use COS Method /<br>Inverse Transform"]
    F --> E
    E --> G["Key Findings:<br>Explicit Exercise Boundary<br>High Efficiency & Accuracy"]
    G --> H["Outcomes:<br>Generalized Framework for<br>Diffusions, Jumps, & Multidimension"]