American and Bermudan options in currency markets under proportional transaction costs

The pricing and hedging of a general class of options (including American, Bermudan and European options) on multiple assets are studied in the context of currency markets where trading in all assets is subject to proportional transaction costs, and where the existence of a risk-free numéraire is not assumed. Probabilistic dual representations are obtained for the bid and ask prices of such options, together with constructions of hedging strategies, optimal stopping times and approximate martingale representations for both long and short option positions.