Hedging Under Loss Constraints.

Summary We present in this section a direct approach to obtain the hedging price of a contingent claim, in the almost sure sense of super-replication or in the sense of a risk criterion (quantile hedging, expected shortfall, utility indifference). This approach, based on the notion of stochastic target, was initiated by Soner and Touzi [55] for the super-replication criterion, and then extended by Bouchard, Elie and Touzi [10] for the hedging under risk control, see also [8, 13] and [14].