A SURVEY ON AMERICAN OPTIONS: OLD APPROACHES AND NEW TRENDS

Abstract

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European op- tion can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up un- til now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These ap- proaches typically provide numerical or approximate analytic methods to �nd the price and the boundary. Topics included in this survey are early approaches(treesnite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, an- alytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochas- tic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.