On the exercise of American quanto options

American option pricing is an important and engaging area of financial economics, particularly so in the presence of negative interest rates. Quanto options offer major international hedging/investment opportunities. We provide a comprehensive description of the optimal exercise policies associated with American quanto options. We show that a non-standard exercise policy characterized by a double continuation region may be optimal in the presence of non-positive domestic interest rates. We study empirical examples of finite-maturity American quanto options for which a double continuation region surrounding a non-empty early exercise region exists even if the infinite-maturity early exercise region is empty and the value of the infinite-maturity option is unbounded. Under the assumptions underpinning such empirical examples, we carefully characterize the existence, the monotonicity properties and the close-to-maturity behaviour of the upper and lower critical prices.