Abstracts

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 the one given in [2], 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. The arbitrage model P is constructed from a reference arbitrage-free model Q by a non-equivalent change of measure. This procedure is not new and goes back to the construction of the Bessel process by [1]. However, we extend it in two directions. Firstly, from the theoretical point of view, we provide a characterization of the superhedging price of a claim under P in terms of the superhedging price of a related claim under Q. This allows us to characterize the optimal arbitrage profit under P in terms of the superhedging price under Q. Secondly, from the economic point of view, we provide an economic intuition for the new arbitrage model as a model implementing the view of the economic agent concerning the impossibility of certain market events. In other words, if an economic agent considers that a certain event (such as a sovereign default) is impossible, but it is actually priced in the market, our method can be used to construct a new model incorporating this arbitrage opportunity, and to compute the associated optimal arbitrage strategy. We then combine these two ideas to develop several new classes of models with optimal arbitrage, allowing for a clear economic interpretation. We also discuss the issue of robustness of these arbitrages to small transaction costs/small observation errors and show that some of our examples are not fragile in the sense of [3].

[1] Delbaen, F. and Schachermayer, W. (1995). Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab. Theory Related Fields, 102(3):357-366
[2] Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. Ann. Appl. Probab., 20(4):1179-1204.
[3] Guasoni, P. and Rasonyi, M. (2012). Fragility of arbitrage and bubbles in diffusion models. Technical report.