Before revision:   We first get the fundamental theorem of asset pricing (FTAP) using semi-static trading strategies . Using the FTAP result, we further get the dualities for the hedging prices of European and American options.
After revision:  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 semi-static trading strategies . Using the FTAP result, we further get the dualities for the hedging prices of European and American options.