Randomized stopping times and American option pricing with transaction costs (Q2707162)

The paper examines the no-arbitrage pricing of American options in a discrete-time market model where stock trades are subject to proportional transaction costs. The authors characterize the range of option prices that is consistent with no arbitrage and find necessary and sufficient conditions for the absence of arbitrage in the model. The central notion of the paper is the notion of randomized stopping times. It is a nonnegative adapted process whose sum on any path is 1. Randomized stopping times play an essential role in expectation representations of the upper hedging price of American options with proportional transaction costs, and the bulk of the paper consists of demonstrating a variety of representations for the upper hedging price. NEWLINENEWLINENEWLINETwo results in the paper explain why randomized strategies are necessary to represent the upper hedging price in the presence of transaction costs. The first result involves Nash equilibria in game theory, where mixed strategies are often necessary. The authors define a natural game between a seller and a ``devil'' where it turns out that at Nash equilibrium, the seller's ``cost'' and the devil's ``utility'' are both equal to the upper hedging price, and the devil must in general choose a mixture of stopping times. The second result makes a distinction between an oblivious hedge (a single portfolio that hedges against all exercise strategies) and an adaptive hedge (a possibly different portfolio to hedge against each exercise strategy). The lower hedging prices are compared with upper ones. The difference is that with nonzero transaction costs, the inner minimization is over approximate martingale measures. Various conditions of absence of arbitrage are proven. The techniques that are used depend on strong duality of linear programming. NEWLINENEWLINENEWLINEAn extended bibliography in the field of market models with transaction costs is presented.