Martingales and Arbitrage: a New Look

This paper addresses the equivalence between the absence of arbitrage and the existence of equivalent martingale measures. The equivalence will be established under quite weak assumptions since there are no conditions on the set of trading dates (it may be finite or countable, with bounded or unbounded horizon, etc.) or on the trajectories of the price process (for instance, they do not have to be right-continuous). Besides we will deal with arbitrage portfolios rather than free-lunches. The concept of arbitrage is much more intuitive than the concept of free lunch and has more clear economic interpretation. Furthermore it is more easily tested in theoretical models or practical applications. In order to overcome the usual mathematical difficulties arising when dealing with arbitrage strategies, the set of states of nature will be widened by drawing on projective systems of Radon probability measures, whose projective limit will be the martingale measure. The existence of densities between the "real" probabilities and the "risk-neutral" probabilities will be guaranteed by introducing the concept of "projective equivalence". Hence some classical counter-examples will be solved and a complete characterization of the absence of arbitrage will be provided in a very general framework.

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