An inverse problem in the pricing of American options is considered. The problem is formulated as an output least-squares problem governed by a parabolic variational inequality in non-ivergence form. The existence of an optimal solution is proved, and first-order optimality conditions of C-stationarity-type are derived by using a relaxation-penalization technique. Numerically, the discrete optimality system is solved by an active-set-Newton solver with feasibility restoration.