Regularity of the American

Regularity of the American

Put option in the Black-Scholes model with general discrete dividends Summary. We analyze the regularity of the value function and of the optimal exercise boundary of the American Put option when the underlying asset pays a discrete dividend at known times during the lifetime of the option. The ex dividend asset price process is assumed to follow the Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date.

This function is assumed to be non-negative, non decreasing and with growth rate not greater than 1. 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.

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 ≤ tI d < tI−1 d < ··· < ti d < ··· < t1 d < T. At each dividend time ti d, the value of the stock becomes Sti d = Sti d− − Di Sti d− where Di(Sti d−) is the value of the dividend payment (see Figure 3.1).

We suppose that each dividend function Di : R+ → R+ is non-decreasing, non-negative and such that x 7→ x−Di(x) is also non-decreasing and non-negative. We are interested in the value of the American Put option with strike K and maturity T. Since we are in a Markovian framework, the price can be characterized in terms of a value function depending of the time t and the stock price at time t. For the sake of consistency, we will denote this value function by u0 for the case without dividends

 We prove that the exercise boundary is continuous at any time which is not a dividend date and that the smooth contact property holds for the value function of the option. 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. continuity) of the exercise boundary was proved to hold globally (resp. 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. In the first section, we introduce our notations and assumptions. In the second section, we recall the existence results for the value function and the exercise boundary stated in [JV11].

The third section is devoted to the smooth-fit property and relies on a viscosity solution approach combined with an estimation of the derivative of the value function with respect to the time variable. In the fourth section, we prove the continuity result for the exercise boundary, which is known to be upper semicontinuous by continuity of the value function. The right-continuity is obtained by comparison with the optimal boundary of the Put option in the Black-Scholes model without dividend.

The left continuity follows from the characterization of the continuation region as the set of points where the spatial derivative of the value function is greater than −1. In the fifth section, we are interested in the local behaviour of the exercise boundary in a neighborhood of the dividend date.

To be able to analyse this behaviour, we have to assume that the stock level at which the dividend function becomes positive lies in the post-dividend exercise region. When the dividend function has a positive slope at this point, we obtain a first order expansion for the exercise boundary at the dividend date. We also provide sufficient conditions for the exercise boundary to be locally monotonic