The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options

TL;DR

This paper investigates how non-smooth payoffs, such as kinks in American options, affect the accuracy and convergence rates of penalty approximation methods, using asymptotic analysis and viscosity solutions.

Contribution

It provides a detailed analysis of the impact of payoff discontinuities on penalty approximation convergence rates and introduces methods to improve numerical accuracy.

Findings

Sharp convergence rates depend on payoff smoothness.

Matched asymptotic expansions characterize boundary layers.

Application of viscosity theory yields bounds on option values.

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. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e., the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. This effect becomes important not least when using penalisation as a numerical technique. We use matched asymptotic expansions to characterise 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 [Jakobsen, 2006] can be applied to this setting to derive upper and lower bounds on the value. In a small extension to [Bensoussan & Lions, 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.