The Stochastic Balance Equation for the American Option Value Function and its Gradient

TL;DR

This paper introduces a stochastic balance equation for valuing and hedging American options on dividend-paying assets within a multidimensional diffusion framework, establishing uniqueness of the solution.

Contribution

It derives a novel stochastic balance equation for American option values and their gradients, proving the solution's uniqueness in this context.

Findings

Established a stochastic balance equation for American options

Proved the uniqueness of the solution to the equation

Applied the framework to multidimensional diffusion models

Abstract

In the paper we consider the problem of valuation and hedging of American options written on dividend-paying assets whose price dynamics follow the multidimensional diffusion model. We derive a stochastic balance equation for the American option value function and its gradient. We prove that the latter pair is the unique solution of the stochastic balance equation as a result of the uniqueness in the related adapted future-supremum problem.