The stochastic balance equation for the American option value function and its gradient

Abstract

In the paper we consider the problem of valuation and hedging of American options written on dividend-paying assets whose price dynamics follow a 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. The latter problem has an attractive interpretation: the given adapted stochastic process can be adjusted by a martingale in such a manner, that the observer will gain the perfect foresight of the resulting future-supremum process via the Snell envelope of the given stochastic process.