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

2025-06-22

AI Summary

Pricing American options under time-varying jump-diffusion models faces three key challenges: the implicit nature of the optimal exercise boundary, the absence of closed-form expressions for the transition density, and complex asymptotic behavior near maturity. Method: This paper proposes a semi-analytical approach grounded in a second-kind Volterra integral equation and characteristic-function-based techniques (e.g., COS expansion). By generalizing decomposition methods to general Lévy processes and multidimensional diffusion frameworks, it derives an explicit integral representation of the optimal exercise boundary—bypassing reliance on closed-form density solutions. Contribution/Results: The method achieves both analytical accuracy and numerical efficiency. Numerical experiments demonstrate substantial improvements in accuracy and computational speed over finite-difference methods and Monte Carlo simulation. It is scalable and robust, enabling industrial-grade large-scale American option pricing.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Security and Privacy: Large-scale security measurements

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.

Problem

• Efficient valuation of American options in time-dependent jump-diffusion models
• Extending decomposition method to general jump-diffusion and Levy models
• Handling cases without closed-form transition densities using characteristic functions

Innovation

• Semi-analytical approach for American options pricing
• Volterra integral equations for exercise boundary
• Characteristic functions for transition densities