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

TL;DR

This paper develops a semi-analytical integral equation method for efficiently valuing American options in complex time-dependent jump-diffusion models, improving over traditional numerical techniques.

Contribution

It extends the decomposition approach to jump-diffusion and Levy models, utilizing characteristic functions for models without closed-form densities, and generalizes to multidimensional cases.

Findings

Method outperforms finite-difference and Monte Carlo methods in efficiency

Successfully handles models without closed-form transition densities

Demonstrates robustness and scalability in numerical examples

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.