  We analyse the obstacle problem for the nonlocal parabolic operator
\[\partial_t u + (-\Delta)^{s} u - b \cdot \nabla u - \mathcal{I}u - ru,\]
where $b\in\mathbb{R}^n$, $r\in\mathbb{R}$, and $\mathcal{I}$ is a nonlocal
lower order diffusion operator with respect to the fractional Laplace operator
$(-\Delta)^{s}$. This model appears in the study of American options pricing
when the stochastic process governing the stock price is assumed to be a purely
jump process. We study the existence and the uniqueness of solutions to the
obstacle problem, and we prove optimal regularity of solutions in space, and
almost optimal regularity in time.
