Approximation of American Put Prices by European Prices via an Embedding Method

Abstract

In mathematical finance, the price of the so-called “American Put option” is given by the value function of the optimal-stopping problem with the option payoff $\psi: x \to (K - x)^+$ as a reward function. Even in the Black–Scholes model, no closed-formula is known and numerous numerical approximation methods have been specifically designed for this problem.

In this paper, as an application of the theoretical result of B. Jourdain and C. Martini [Ann. Inst. Henri Poincaré Anal. Nonlinear 18 (2001) 1–17], we explore a new approximation scheme: we look for payoffs as close as possible to $\psi$, the American price of which is given by the European price of another claim. We exhibit a family of payoffs $\hat{\varphi}_h$ indexed by a measure $h$, which are continuous, match with $(K - x)^+$ outside of the range $]K_*, K[$ (where $K_*$ is the perpetual Put strike), are analytic inside with the right derivative ( -1) at both ends. Moreover a numerical procedure to select the best $h$ in some sense yields nice results.