On the Exercise of American quanto Options

American quanto options are derivatives written on a foreign risky asset that can be exercised by the holder during the whole life of the contract. We provide a comprehensive description of the optimal exercise policies associated with American quanto plain vanilla options. In particular, 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. In fact, it may be optimal to early exercise American quanto options if they are enough in the money but not if they are too deeply in the money. It is the interplay of domestic and foreign interest rates with dividend rate, volatility of the foreign risky asset as well as its correlation with the exchange rate that makes non-standard deeply in the money continuation region appear. 

We study empirical examples of finite-maturity American quanto put 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. For such empirical examples, we carefully characterize the existence, the monotonicity properties and the asymptotics at maturity of the free boundaries separating the (single or double) continuation region and the early exercise region.