Appl Math Optim 35:145–164 (1997) © 1997 Springer-Verlag New York Inc. Optimal Stopping, Free Boundary, and American Option in a Jump-Diffusion Model Huyˆ en Pham ∗ CEREMADE, Universit´ e Paris IX Dauphine, Place du Mar´ echal de Lattre de Tassigny, 75775 Paris Cedex, France and CREST, Laboratoire de Finance-Assurance, 15 boulevard Gabriel P´ eri, 92245 Malakoff Cedex, France Communicated by A. Bensoussan Abstract. This paper considers the American put option valuation in a jump- diffusion model and relates this optimal-stopping problem to a parabolic integro- differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black–Scholes (BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary and also give one estimate near maturity, generalizing a recent result of Barles et al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American put option price and its critical stock price. Key Words. Optimal stopping, Jump-diffusion model, American option, Free- boundary problem. AMS Classification. 60G40, 90A09, 93E20. ∗ Now affiliated to Equipe d’Analyse et de Math´ ematiques Appliqu´ ees, Universit´ e de Marne la Vall´ ee, 2 rue de la Butte verte, 93166 Noisy le Grand cedex, France.