Title: Semi-analytical pricing of American options with hybrid dividends via integral equations and the GIT method Show Chinese title

Title: 基于积分方程和GIT方法的混合股息美式期权半解析定价

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. Show Chinese abstract

Abstract: 本文介绍了一种半解析方法，用于定价支付离散和/或连续股息的资产（股票、ETF）的美式期权。 这个问题因其复杂性而著称，因为离散股息会导致价格突然下跌并影响最优行权时间，使得传统的连续股息模型不适用。 我们的方法利用了作者及其合作者在多篇论文中提出的广义积分变换（GIT）方法，该方法将带有自由边界的复杂偏微分方程定价问题转化为积分伏尔泰拉方程的第二类或第一类。 在本文中，我们通过考虑一个流行的几何布朗运动（GBM）模型来说明这种方法，该模型使用狄拉克δ函数来考虑离散现金股息和比例股息。 通过将问题重新表述为积分方程，我们可以依次求解期权价格和提前行权边界，从而有效处理由股息引起的不连续性。 我们的方法为标准数值技术（如二叉树或有限差分方法）提供了一种强大的替代方案，这些方法在处理离散股息的跳跃条件时可能会失去精度或性能。 多个示例表明，GIT方法具有高度的准确性且计算效率高，无需依赖广泛的计算网格或复杂的向后归纳步骤。