The paper deals with construction of a hedging strategy which maximizes the probability of a successful hedge under the objective measure \(P\), given a constraint on the required cost. This concept of quantile hedging can be considered as a dynamic version of the well-known value at risk concept. First the authors consider the general case of complete market. They determine a set of maximal probability under the constraint that the cost of hedging the given claim on that set satisfies a given bound. Using the Neyman-Pearson lemma, this set is constructed as an optimal test where the alternative is given by the objective measure \(P\), and where the hypothesis is defined in terms of the contingent claim and the equivalent martingale measure \(P^{*}\). Then authors use the completeness of model in order to replicate the knockout option obtained by restricting the claim to this maximal set. This strategy maximizes the probability of a successful hedge. The general incomplete case also is considered. The technique of superhedging is used. As an illustrations of this approach the authors compute the strategy of quantile hedging for call option in different models for the price fluctuation of the underlying asset.