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Abstract
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We will discuss the free boundary problem associated to the American option pricing problem for jump diffusions. Because of the jump in the underlying process, this problem is a free boundary problem for parabolic integro-differential equations. Using the tools in proving regularity results for parabolic differential equations and especially the Stefan problem, we will show that the optimal exercise boundary/ free boundary of the American option is continuously differentiable (except at the maturity). Moreover, we will upgrade its regularity and show it is infinitely differentiable under an appropriate regularity assumption on the jump distribution. Our regularity result improves the result of Yang, Jiang and Bian in 2006 by removing a technical assumption on the parameters.
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