Abstract: Collar option is designed for invertors with a downside
income, and for issuers with a limit loss. The mathematical pricing
model of the American collar option can be formulated as a
one-dimensional parabolic variational inequality, or equivalently, a
free boundary problem. To solve this problem, the penalty method and
PDE arguments are applied. The existence and uniqueness of the
solution, the properties of the free boundaries, such as
monotonicity, smoothness, and location, are presented.

Key words: American collar option, option pricing, optimal exercise boundary