Semi-analytical pricing of American options with hybrid dividends via integral equations and the GIT method

ArXiv ID: 2510.18159 “View on arXiv”

Authors: Andrey Itkin

Abstract

This paper introduces a semi-analytical method for pricing American options on assets (stocks, ETFs) that pay discrete and/or continuous dividends. The problem is notoriously complex because discrete dividends create abrupt price drops and affect the optimal exercise timing, making traditional continuous-dividend models unsuitable. Our approach utilizes the Generalized Integral Transform (GIT) method introduced by the author and his co-authors in a number of papers, which transforms the pricing problem from a complex partial differential equation with a free boundary into an integral Volterra equation of the second or first kind. In this paper we illustrate this approach by considering a popular GBM model that accounts for discrete cash and proportional dividends using Dirac delta functions. By reframing the problem as an integral equation, we can sequentially solve for the option price and the early exercise boundary, effectively handling the discontinuities caused by the dividends. Our methodology provides a powerful alternative to standard numerical techniques like binomial trees or finite difference methods, which can struggle with the jump conditions of discrete dividends by losing accuracy or performance. Several examples demonstrate that the GIT method is highly accurate and computationally efficient, bypassing the need for extensive computational grids or complex backward induction steps.

Keywords: American Options, Generalized Integral Transform (GIT), Discrete Dividends, Option Pricing, Free Boundary Problem, Equities

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is highly mathematically dense, featuring advanced transformations like the Generalized Integral Transform and solving Volterra integral equations with singularities. However, it lacks empirical validation through backtesting, code implementations, or statistical performance metrics, focusing instead on theoretical derivation and numerical experiments.

  flowchart TD
    A["Research Goal: Price American Options with Hybrid Dividends"] --> B["Model: GBM with Discrete Cash & Proportional Dividends"]
    B --> C["Apply GIT Method"]
    C --> D["Transform PDE with Free Boundary"]
    D --> E["Solve Integral Volterra Equation"]
    E --> F{"Compute Price & Early Exercise Boundary"}
    F --> G["Outcomes: High Accuracy & Computational Efficiency"]
    G --> H["Comparison: Superior to Binomial/Finite Difference Methods"]
    style A fill:#e1f5fe
    style G fill:#e8f5e8
    style H fill:#e8f5e8