EFFICIENT HEDGING OF EUROPEAN OPTIONS WITH ROBUST CONVEX LOSS FUNCTIONALS: A DUAL-REPRESENTATION FORMULA

Abstract

Motivated by numerical representations of robust utility functionals, due to Maccheroni et al., we study the problem of partially hedging a European option H when a hedging strategy is selected through a robust convex loss functional L(·) involving a penalization term γ(·) and a class of absolutely continuous probability measures . We present three results. An optimization problem is defined in a space of stochastic integrals with value function EH(·). Extending the method of Föllmer and Leukerte, it is shown how to construct an optimal strategy. The optimization problem EH(·) as criterion to select a hedge, is of a “minimax” type. In the second, and main result of this paper, a dual-representation formula for this value is presented, which is of a “maxmax” type. This leads us to a dual optimization problem. In the third result of this paper, we apply some key arguments in the robust convex-duality theory developed by Schied to construct optimal solutions to the dual problem, if the loss functional L(·) has an associated convex risk measure ρL(·) which is continuous from below, and if the European option H is essentially bounded.