The Obstacle Problem Arising from the American Chooser Option

Abstract

We study the finite-horizon obstacle problem associated with the American chooser option. Its obstacle is the maximum of two independently determined American call and put values and has a spatial kink where these values cross. Using truncation, penalization, and comparison, we construct a local strong Sobolev solution and prove uniqueness within an exponential-growth class. We show that contact occurs only on the smooth exercise-payoff branches and that both contact regions have nonempty sections at every time. We then establish strict monotonicity, terminal limits, and local Lipschitz continuity of the two free boundaries. Localized one-phase equations, quantitative nondegeneracy, and parabolic boundary Harnack estimates yield $C^{\infty }$ regularity on the open time interval.

33 pages