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. 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 omnipresent `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 jumpdiffusion 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 jumpdiffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.
 Publication:

arXiv eprints
 Pub Date:
 August 2010
 arXiv:
 arXiv:1008.0836
 Bibcode:
 2010arXiv1008.0836H
 Keywords:

 Quantitative Finance  Computational Finance;
 Mathematics  Functional Analysis
 EPrint:
 34 Pages, 10 Figures