The Effect of Nonsmooth Payoffs on the Penalty Approximation of American Options

Abstract

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. We use matched asymptotic expansions to characterize the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [E. R. Jakobsen, Asymptot. Anal., 49 (2006), pp. 249--273] can be applied to derive upper and lower bounds on the option value. This analysis confirms the higher order of accuracy in the penalty parameter for convex payoffs (compared to the general case) seen earlier in numerical tests and from asymptotic expansions. In a small extension to [A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control, Stud. Math. Appl. 12, North-Holland, Amsterdam, New York, Oxford, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.