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

2025-10-20

AI Summary

Pricing American options under mixed dividends—discrete (lump-sum or proportional) and continuous—is challenging because conventional continuous-dividend models fail to capture dividend-induced price jumps and the resulting discontinuities in the optimal exercise boundary. Method: This paper proposes a semi-analytical approach based on the Generalized Integral Transform (GIT), the first application of GIT to mixed-dividend settings. It transforms the free-boundary PDE into a second-kind (or first-kind) Volterra integral equation, enabling sequential, efficient computation of both the exercise boundary and option price. The underlying asset follows geometric Brownian motion, with discrete dividends modeled precisely via Dirac delta functions. Results: Numerical experiments demonstrate high accuracy and computational efficiency; the method significantly outperforms binomial trees and finite-difference schemes in stability and cost when handling dividend-induced jumps.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Pricing American options with hybrid dividends
• Handling discontinuities from discrete dividends
• Transforming PDE into solvable integral equations

Innovation

• Semi-analytical American option pricing with hybrid dividends
• Transforms PDE to Volterra equation via GIT method
• Handles dividend discontinuities using Dirac delta functions