The Obstacle Problem Arising from the American Chooser Option

TL;DR

This paper analyzes a complex obstacle problem from American chooser options, establishing existence, uniqueness, and properties of the solution and free boundary through rigorous theoretical analysis.

Contribution

It introduces a novel approach to solving the obstacle problem for American chooser options, proving key properties of the solution and free boundary.

Findings

Existence and uniqueness of a strong solution established

Monotonicity and smoothness of the free boundary proven

The structure of the obstacle problem is characterized in detail

Abstract

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.