Robust efficient hedging for American options: The existence of worst case probability measures

Our problem here starts when the seller of an American option H has an initial capital c and aims to control shortfall risk by optimally trading in the market. We consider the case where the initial capital c is smaller than the superhedging cost of H. Thus, it is meaningful to consider efficient strategies of partial hedging. We investigate in the American case the approach of Föllmer and Leukert of efficiently hedging European options. At the same time, we consider situations where the model R is ambiguous. This implies that the seller´s preference is going to be represented by a robust loss functional, involving a whole class of absolutely continuous probability measures Q. We apply a construction due to Föllmer and Kramkov on the existence of Fatou-convergent sequences of convex combinations of supermartingales, to show that optimal strategies minimizing such a functional exist. We then show that the robust efficient hedging problem can be reduced into a non-robust problem of efficient hedging with respect to a worst case probability measure Q*∈Q, if Q satisfies a compactness condition and H is essentially bounded.