Free boundary problem of a new type of American Option

A parabolic variational inequality with two free boundaries arising from a new type of stock option pricing is considered, which provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low. The existence and uniqueness of solution to the problem via the PDE technique are proved. Moreover, the existence, monotonicity and smoothness of two free boundaries, and the relationship between the solution to the parabolic variational inequality and the guaranteed minimum are obtained.