  In this paper we provide a quantitative analysis to the concept of arbitrage,
that allows to deal with model uncertainty without imposing the no-arbitrage
condition. In markets that admit ``small arbitrage", we can still make sense of
the problems of pricing and hedging. The pricing measures here will be such
that asset price processes are close to being martingales, and the hedging
strategies will need to cover some additional cost. We show a quantitative
version of the Fundamental Theorem of Asset Pricing and of the
Super-Replication Theorem. Finally, we study robustness of the amount of
arbitrage and existence of respective pricing measures, showing stability of
these concepts with respect to a strong adapted Wasserstein distance.
