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

TL;DR

This paper presents a semi-analytical GIT-based method for pricing American options with discrete and continuous dividends, transforming the problem into an integral equation to improve accuracy and efficiency over traditional numerical methods.

Contribution

The paper introduces a novel application of the GIT method to handle American options with complex dividend structures, providing a more accurate and efficient alternative to existing techniques.

Findings

GIT method accurately prices American options with dividends.

The approach outperforms traditional numerical methods in efficiency.

It effectively manages discontinuities caused by dividends.

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.