Martingales and Fuzzy Stopping Times in the Pricing of American Options with Transaction Costs

In a general discrete-time market model with proportional transaction costs, we derive new expectation representations of the range of arbitrage-free prices of an arbitrary American option. A surprising feature of the upper bound representation is that it necessarily involves a randomized exercise strategy, given by a mixture of ordinary stopping times. In particular the upper bound is the maximum expected discounted payoff achievable, over randomized exercise strategies, and probability measures that are nearly martingale-measures. The lower bound expression is similar, except that the maximization is over ordinary exercise strategies, and the minimum over measures is taken, rather than the maximum. We also show that a necessary and sufficient condition for the absence of arbitrage is the existence of a certain type of nearly-martingale measure. Our results make use of strong duality in linear programming.