  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.
