We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian. This problem arises in American option models when the assets prices are driven by pure jump L évy processes.

Our main result establishes that, when $s>1/2$, the free boundary is a $C^{1,\alpha}$ graph in $x$ and $t$ near any regular free boundary point. Furthermore, we also prove that solutions are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion.