Market Models with Optimal Arbitrage

We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zero-interest rate setting, starting with an initial wealth of 1 and using only positive portfolios, to superreplicate a constant $c>1$. The optimal arbitrage strategy is the strategy for which this constant has the highest possible value. Our definition of optimal arbitrage is similar to that in [D. Fernholz and I. Karatzas, Ann. Appl. Probab., 20 (2010), pp. 1179--1204], where optimal relative arbitrage with respect to the market portfolio is studied. In this work we present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly. We then develop several new examples of market models with arbitrage, which are based on economic agents' views concerning the impossibility of certain events rather than ad hoc constructions. We also explore the robustness of arbitrage strategies with respect to small perturbations of the price process and provide new examples of arbitrage models which are robust in this sense.