  The parabolic obstacle problem for the fractional Laplacian naturally arises
in American option models when the assets prices are driven by pure jump L\'evy
processes. In this paper we study the regularity of the free boundary. Our main
result establishes that, when $s>\frac12$, the free boundary is a
$C^{1,\alpha}$ graph in $x$ and $t$ near any regular free boundary point
$(x_0,t_0)\in \partial\{u>\varphi\}$. Furthermore, we also prove that solutions
$u$ are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion of
the form \[u(x,t)-\varphi(x)=c_0\bigl((x-x_0)\cdot
e+a(t-t_0)\bigr)_+^{1+s}+o\bigl(|x-x_0|^{1+s+\alpha}+
|t-t_0|^{1+s+\alpha}\bigr),\] with $c_0>0$, $e\in \mathbb{S}^{n-1}$, and $a>0$.
