The Obstacle Problem Arising from the American Chooser Option

We study the obstacle problem associated with the American chooser option. The obstacle is given by the maximum of an American call option and an American put option, which, in turn, can be expressed as the maximum of the solutions to the corresponding obstacle problems. This structure makes the obstacle problem particularly challenging and non-trivial. Using theoretical analysis, we overcome these difficulties and establish the existence and uniqueness of a strong solution. Furthermore, we rigorously prove the monotonicity and smoothness of the free boundary arising from the obstacle problem.