On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing

The following paper is devoted to the study of the positivity set $U=\{\mathcal{L}\phi>0\}$ arising in parabolic obstacle problems. It is shown that $U$ is contained in the non-coincidence set with a positive distance between the boundaries uniformly in the spatial variable if the boundary of $U$ satisfies an interior $C^1$-Dini condition in the space variable and a Lipschitz condition in the time variable. We apply our results to American option pricing and we thus show that the positivity set is strictly contained in the continuation region, which means that the option should not be exercised in $U$ or on the boundary of $U$.