Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends

Abstract

We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time t d during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterised in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, may no longer be monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary is non-increasing in a left-hand neighbourhood of t d , and tends to 0 as time tends to t − d with a speed that we can characterize. When the dividend function is linear in a neighbourhood of zero, then we show continuity of the exercise boundary and a high contact principle in the left-hand neighbourhood of t d . When it is globally linear, then right-continuity of the boundary and the high contact principle are proved to hold globally. Finally, we show how all the previous results can be extended to multiple dividend payment dates in that case.

Regularity of the American Put option in the Black–Scholes model with general discrete dividends

“…We prove that the exercise boundary is continuous and that the smooth contact property holds for the value function at any time but the dividend dates. We thus extend and generalize the results obtained in [JV11] when the dividend function is also positive and concave. Lastly, we give conditions on the dividend function ensuring that the exercise boundary is locally monotonic in a neighborhood of the corresponding dividend date.…”

“…in a left-hand neighborhood of each dividend date). We also extend the result obtained in [JV11] on the decrease of the exercise boundary in a left-hand neighborhood of the first (resp of each) dividend date when the corresponding dividend function was assumed to be positive and concave (resp. when all dividend functions were supposed to be linear) : we give more general sufficient conditions on each dividend function for the exercise boundary to be either non-decreasing or non-increasing in a left-hand neighborhood of the corresponding dividend date.…”

“…We considerably extend the results obtained in [JV11], where the continuity of the exercise boundary and the smooth contact property were only obtained in a left-hand neighborhood of the first dividend date when the corresponding dividend function was assumed to be globally concave and linear with a positive slope in a neighbohood of the origin. Under the much more restrictive assumption of global linearity of all the dividend functions, the smooth contact property and the right-continuity (resp.…”

“…Integral formulas for the exercise boundary which are similar to the ones in [CJM92] have been derived under the assumption that the boundary is Lipschitz continuous (see Göttsche and Vellekoop [GV11]) or locally monotonic (Vellekoop & Nieuwenhuis [VNar]). In this paper we continue the study, undertaken in [JV11], of conditions under which such regularity properties of the optimal exercise boundary under discrete dividend payments can be proven.…”

“…We consider the American put option with maturity T and strike K written on an underlying stock S. Like in [JV11], we assume that the stochastic dynamics of the ex-dividend price process of this stock can be modelled by the Black Scholes model and that this stock is paying discrete dividends at deterministic times 0…”