Semianalytical Pricing of American Options with Hybrid Dividends via Integral Equations and the GIT Method

This article introduces a semianalytical method for pricing American options on assets (stocks, ETFs) that pay discrete 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 uses the Generalized Integral Transform (GIT) method introduced by the author and his co-authors in several previous publications, 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 article, 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 such as 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, avoiding the need for extensive computational grids or complex backward induction steps.