Abstract: Motivated by some pricing models of American options for baskets, we consider a parabolic variational inequality with a non-local operator and a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We propose an a posteriori error estimator and show that it localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also discuss fast solvers for the discrete variational inequalities on adaptive grids. Finally, we show that the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions by some numerical experiments.