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

## Author

Listed:
• Benjamin Jourdain

() (CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique - Inria - Institut National de Recherche en Informatique et en Automatique - ENPC - École des Ponts ParisTech, MATHRISK - Mathematical Risk handling - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech)

• Michel Vellekoop

(ASE - Amsterdam School of Economics - UvA - University of Amsterdam [Amsterdam])

## 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.

## Suggested Citation

• Benjamin Jourdain & Michel Vellekoop, 2009. "Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends," Working Papers hal-00436327, HAL.
• Handle: RePEc:hal:wpaper:hal-00436327
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00436327v2
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File URL: https://hal.archives-ouvertes.fr/hal-00436327v2/document

## References listed on IDEAS

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1. Fudenberg, Drew & Imhof, Lorens A., 2008. "Monotone imitation dynamics in large populations," Journal of Economic Theory, Elsevier, vol. 140(1), pages 229-245, May.
2. Michel BenaÔm & J–rgen W. Weibull, 2003. "Deterministic Approximation of Stochastic Evolution in Games," Econometrica, Econometric Society, vol. 71(3), pages 873-903, May.
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