Abstract: By appplying the comparison principle for the variational
inequality, we analyzed the behavior of exercise boundaries for the
perpetual American straddle option. We found that it is a free
boundary problem. Different from the standard perpetual American
option, it has two exercise boundary points with dividends and only
one free boundary point without dividends. These results can be
understood very well from the financial point of view. We will
present a rigorous mathematical proof, and find the bounds of
exercise boundaries for the American straddle option with finite
expiry.

Key words: straddle option, exercise boundary, variational inequality