Hedging of options with a given probability

We consider a model of a complete market with two assets under the suggestion that an investor may hedge the payoff function with the given probability; in other words, the investor should have capital not less than the given payoff function with probability not less than $1-\alpha$ ($\alpha $ is a given significance level). Under some limitations on a class of hedging strategies we find a lower bound for an option price (that is, for the initial capital of the investor) and construct a hedge (the investor strategy) for which this lower bound is achieved. For examples, we calculate the price and hedge of a European call option and also an American call option with a barrier condition.