  We re-examine and extend the findings from the recent paper by Dumitrescu,
Quenez and Sulem (2018) who studied American and game options in a particular
market model using the nonlinear arbitrage-free pricing approach developed in
El Karoui and Quenez (1997). In the first part, we provide a detailed study of
unilateral valuation problems for the two counterparties in an American-style
contract within the framework of a general nonlinear market. We extend results
from Bielecki and Rutkowski (2015) and Bielecki, Cialenco and Rutkowski (2018)
who examined the case of a European-style contract. In the second part, we
present a BSDE approach, which is used to establish more explicit pricing,
hedging and exercising results when solutions to reflected BSDEs have
additional desirable properties.
