Super-hedging a European option with a coherent risk-measure and without no-arbitrage condition

In this paper, we revisit the discrete-time super hedging problem of contingent claims with respect to a dynamic risk-measure defined by its acceptance sets. Without any no-arbitrage condition, we show that it is possible to characterize the prices of an European claim. Our analysis reveals a natural weak no-arbitrage condition that we study. This is a condition formulated for the prices instead of the attainable claims. Our approach is not based on a robust representation of the risk-measure and we do not suppose the existence of a risk-neutral probability measure.