  We consider a financial market where stocks are available for dynamic
trading, and European and American options are available for static trading
(semi-static trading strategies). We assume that the American options are
infinitely divisible, and can only be bought but not sold. In the first part of
the paper, we work within the framework without model ambiguity. We first get
the fundamental theorem of asset pricing (FTAP). Using the FTAP, we get the
dualities for the hedging prices of European and American options. Based on the
hedging dualities, we also get the duality for the utility maximization. In the
second part of the paper, we consider the market which admits non-dominated
model uncertainty. We first establish the hedging result, and then using the
hedging duality we further get the FTAP. Due to the technical difficulty
stemming from the non-dominancy of the probability measure set, we use a
discretization technique and apply the minimax theorem.
