*Accepted Paper*

**Inserted:** 26 jun 2017

**Last Updated:** 26 jun 2017

**Journal:** Comm. Pure Appl. Math.

**Year:** 2017

**Abstract:**

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+\kappa(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 $a>0$.

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