Partial hedging of American contingent claims in a finite discrete time model
The shortfall risk minimization problem for the investor who hedges an American contingent claim is studied. The lower bound for the minimal shortfall risk is obtained by maximizing some function over the set all randomized stopping times. It is proved that in a binomial model this lower bound is equal to the minimal shortfall risk. An example is given where the maximum of the function considered over all pure stopping times is less than the minimal shortfall risk. It is shown that the optimal strategy in a binomial model is obtained by superhedging a contingent claim connected with a randomized stopping time which is a solution of an auxiliary maximization problem. There is a similarity of the results obtained to those for European options.