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 Q. 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 ρ L (·) which is continuous from below, and if the European option H is essentially bounded.
Abstract. The numerical representation of convex risk measures beyond essentially bounded financial positions is an important topic which has been the theme of recent literature. In other direction, it has been discussed the assessment of essentially bounded risks taking explicitly new information into account, i.e., conditional convex risk measures. In this paper we combine these two lines of research. We discuss the numerical representation of conditional convex risk measures which are defined in a space L p (F, R), for p ≥ 1, and take values in L 1 (G, R) (in this sense, real-valued). We show how to characterize such a class of real-valued conditional convex risk measures. In the first result of the paper, we see that real-valued conditional convex risk measures always admit a numerical representation in terms of a nice class of "locally equivalent" probability measures Q q,∞ e,loc . To this end, we use the recent extended Namioka-Klee Theorem, due to Biagini and Frittelli. The second result of the paper says that a conditional convex risk measure defined in a space L p (F, R) is real-valued if and only if the corresponding minimal penalty function satisfies a coerciveness property, as introduced by Cheridito and Li in the non-conditional case. This characterization, together with an invariance property will allow us to characterize conditional convex risk measures defined in a space L ∞
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