In this paper we derive robust super-and subhedging dualities for contingent claims that can depend on several underlying assets. In addition to strict super-and subhedging, we also consider relaxed versions which, instead of eliminating the shortfall risk completely, aim to reduce it to an acceptable level. This yields robust price bounds with tighter spreads. As examples we study strict superand subhedging with general convex transaction costs and trading constraints as well as risk-based hedging with respect to robust versions of the average value at risk and entropic risk measure. Our approach is based on representation results for increasing convex functionals and allows for general financial market structures. As a side result it yields a robust version of the fundamental theorem of asset pricing.2010 Mathematics Subject Classification: 91G20, 46E05, 60G42, 60G48

We give a dual representation of minimal supersolutions of BSDEs with non-bounded, but integrable terminal conditions and under weak requirements on the generator which is allowed to depend on the value process of the equation. Conversely, we show that any dynamic risk measure satisfying such a dual representation stems from a BSDE. We also give a condition under which a supersolution of a BSDE is even a solution.framework of Drapeau et al. [6]. The H 1 -L ∞ duality turns out to be the right candidate to constitute the basis of our representation. As a starting point, we consider the set of essentially bounded terminal conditions. In this case, we obtain a dual representation of the minimal supersolution at time 0 and a pointwise robust representation in the dynamic case. We show that when the generator of the equation is decreasing in the value process, the minimal supersolution defines a time consistent cash-subadditive risk measure. It allows for a dual representation on the space of essentially bounded random variables, which agrees with the representation of El Karoui and Ravanelli [8] obtained for BSDE solutions. Our dual representation is obtained by showing that the representation of El Karoui and Ravanelli [8] can be restricted on a smaller set. Then we can use truncation and approximation arguments to obtain the representation in the general case, due to monotone stability of minimal supersolutions. A direct consequence of our representation is the identification of BSDEs solution and minimal supersolution in the case of linear growth generators. Note that our truncation technique appears already in the work of Delbaen et al. [4] where it is used to construct a sequence of µ-dominated risk measures. Furthermore, prior to us Barrieu and El Karoui [1] and Bion-Nadal [2] already used the BM O-martingale theory in the study of financial risk measures, but in different settings from ours. Using standard convex duality arguments such as the Fenchel-Moreau theorem and the properties of the Fenchel-Legendre transform of a convex functional, we extend our dual representation to the set of random variables that can be identified to H 1 -martingales. Notice that this representation is obtained in the static case. Our representation results can be seen as extensions of the dual representation of the minimal superreplicating cost of El Karoui and Quenez [7] to the case where we allow for a nonlinear cost function in the dynamics of the wealth process. The second theme of this work is to give conditions based on convex duality under which a dynamic cash-subadditive risk measure with a given representation can be seen as the solution, or the minimal supersolution of a BSDE. The cash-additive case has been studied by Delbaen et al. [5]. Their results are based on m-stability of the dual space, some supermartingale property and Dood-Meyer decomposition of the risk measure. We shall show that in the cash-subadditive case, discounting the risk measure yields similar results, hence showing an equivalent relationship betwee...

We provide a model-free pricing-hedging duality in continuous time. For a frictionless market consisting of d risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path dependent European option has the same value as the purely probabilistic problem of finding the supremum of the expectations of the option over all martingale measures. The superhedging problem is formulated with simple trading strategies, the claim is the limit inferior of continuous functions, which allows for upper and lower semi-continuous claims, and superhedging is required in the pathwise sense on a σ-compact sample space of price trajectories. If the sample space is stable under stopping, the probabilistic problem reduces to finding the supremum over all martingale measures with compact support. As an application of the general results we deduce dualities for Vovk's outer measure and semi-static superhedging with finitely many securities.MSC 2010: 60G44, 91G20, 91B24.

Accounting for model uncertainty in risk management and option pricing leads to infinite dimensional optimization problems which are both analytically and numerically intractable. In this article we study when this hurdle can be overcome for the so-called optimized certainty equivalent risk measure (OCE) -including the average value-at-risk as a special case. First we focus on the case where the uncertainty is modeled by a nonlinear expectation which penalizes distributions that are "far" in terms of optimaltransport distance (Wasserstein distance for instance) from a given baseline distribution. It turns out that the computation of the robust OCE reduces to a finite dimensional problem, which in some cases can even be solved explicitly. This principle also applies to the shortfall risk measure as well as for the pricing of European options. Further, we derive convex dual representations of the robust OCE for measurable claims without any assumptions on the set of distributions. Finally, we give conditions on the latter set under which the robust average value-at-risk is a tail risk measure. AUTHORS INFO

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