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

Andrey Itkin

TL;DR

This work addresses the challenge of pricing American options on dividend paying assets by combining a decomposition formula with a Generalized Integral Transform (GIT) to convert a free boundary PDE into a Volterra integral equation. The method handles discrete cash and proportional dividends within a time-inhomogeneous Geometric Brownian Motion, using Dirac delta representations for dividends and solving for the early exercise boundary via nonlinear Volterra equations. A European price and the transition density are computed through integral equations with Gaussian kernels, enabling efficient evaluation of the early exercise premium and, consequently, the American price. The approach yields high accuracy and computational efficiency, supports de-Americanization and implied-strike calibration for local volatility surfaces, and demonstrates robust performance across multiple dividend scenarios, including cash, proportional, and continuous dividends. The framework is extendable to other one-factor models and offers practical value for pricing, hedging, and calibration in dividend-rich markets.

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.

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

TL;DR

This work addresses the challenge of pricing American options on dividend paying assets by combining a decomposition formula with a Generalized Integral Transform (GIT) to convert a free boundary PDE into a Volterra integral equation. The method handles discrete cash and proportional dividends within a time-inhomogeneous Geometric Brownian Motion, using Dirac delta representations for dividends and solving for the early exercise boundary via nonlinear Volterra equations. A European price and the transition density are computed through integral equations with Gaussian kernels, enabling efficient evaluation of the early exercise premium and, consequently, the American price. The approach yields high accuracy and computational efficiency, supports de-Americanization and implied-strike calibration for local volatility surfaces, and demonstrates robust performance across multiple dividend scenarios, including cash, proportional, and continuous dividends. The framework is extendable to other one-factor models and offers practical value for pricing, hedging, and calibration in dividend-rich markets.

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.
Paper Structure (32 sections, 1 theorem, 151 equations, 9 figures, 2 tables)

This paper contains 32 sections, 1 theorem, 151 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Compound option approach & Escrowed dividend models.
  3. Lattice-Based Methods (Trees).
  4. Partial differential equation and finite-difference methods.
  5. Monte-Carlo methods.
  6. Recent developments.
  7. Model
  8. The decomposition formula
  9. Computing expectations in \ref{['decompGen']}
  10. The European option price
  11. Transition density function
  12. Computing the exercise boundary $S_B(t)$
  13. The exercise boundary for the American Put option
  14. Singularity in $y'(\tau)$ at $\tau \to 0$
  15. The no-dividends case
  16. ...and 17 more sections

Key Result

proposition 1

Conditional on $S_t = S$, the American Put price with a single exercise boundary $S_B(t)$ can be represented by the following decomposition formula where $\DF(t,s) = e^{-\int_t^s r(u)\,du}$ is the deterministic discount factor, and $\mu(t,S)$ is the drift of the corresponding underlying process. Here, the first term represents the European Put option price $P_E\left(t, S \right)$ while the secon

Figures (9)

  • Figure 1: The difference between the "exact" (computed numerically) and approximating (by \ref{['appr1']}) values of $I_2(x)$ for $\tau = 0.02, 0.125, 0.3$.
  • Figure 2: Early exercise boundaries for an American Put option under the time-homogeneous GBM model: (a) Using the model parameters from Table \ref{['tab1']} with no dividends. The boundary $S_B(t)$ is computed using our method, while $S_F(t)$ is from the binomial tree method. (b) The same test where the EB is computed by solving \ref{['VolEB-a2']} taking into account only the first term under the integral (curve $S_B(t)$) and both terms (curve $S_{B,2}(t)$).
  • Figure 3: Early exercise boundaries for an American Put option under the time-homogeneous GBM model: (a) Using the model parameters from Table \ref{['tab1']} with no dividends. The boundary $S_B(t)$ is computed using our method, while $S_F(t)$ is from the binomial tree method. (b) The same test is repeated with $r = 0.01, \sigma = 0.3$, and a coarse $400 \times 400$ tree grid, which produces a non-monotonic early exercise boundary.
  • Figure 4: Early exercise boundaries for an American Put option under the time-inhomogeneous GBM model: (a) Using the model parameters from Table \ref{['tab2']}. The boundary $S_B(t)$ is computed using our method, while $S_F(t)$ is from the binomial tree method with $500 \times 2000$ points. (b) Time-dependent parameters of the model.
  • Figure 5: Early exercise boundaries for an American Put option under the time-inhomogeneous GBM model. (a) Time-dependent model parameters $r(t)$, $q(t)$, and $\sigma(t)$, accounting for discrete proportional dividends. (b) The EB $S_B(t)$ is computed using our method, while $S_F(t)$ is from a binomial tree method that uses no discrete dividends and constant parameters, obtained by averaging the time-dependent parameters over the option's lifetime.
  • ...and 4 more figures

Theorems & Definitions (3)

  • proposition 1: Proposition 1 in ItkinKitapbayev2025
  • proof : Proof
  • remark 1