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Optimal stopping, free boundary, and American option in a jump-diffusion model. (English) Zbl 0866.60038
Summary: 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) model [see F. Block and M. Scholes, J. Polit. Econ. 81, 637-659 (1973)], 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 G. J. Barles, J. Burdeau, M. Romana and N. Samsoen [Math. Finan. 5, No. 2, 77-95 (1995)] 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.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
91B28 Finance etc. (MSC2000)
93E20 Optimal stochastic control
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