In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class S of significant sets, which we call Arbitrage de la classe S. The choice of S reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular: for S = {Ω} , absence of Model Independent Arbitrage is equivalent to the existence of a martingale measure; for S being the open sets, absence of Open Arbitrage is equivalent to the existence of full support martingale measures. These results are obtained by adopting a technical filtration enlargement and by constructing a universal aggregator of all arbitrage opportunities. We further introduce the notion of market feasibility and provide its characterization via arbitrage conditions. We conclude providing a dual representation of Open Arbitrage in terms of weakly open sets of probability measures, which highlights the robust nature of this concept.

In a model free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semi-static strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path ω ∈ Ω, might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and prove that it is an analytic set.

We develop a robust framework for pricing and hedging of derivative securities in discretetime financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain an abstract (pointwise) Fundamental Theorem of Asset Pricing and Pricing-Hedging Duality. Our results are general and in particular include so-called model independent results of Acciaio et al. (2016); Burzoni et al. (2016) as well as seminal results of Dalang et al. (1990) in a classical probabilistic approach.Our analysis is scenario-based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.

We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean-Vlasov jumpdiffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution.

Recently, financial industry and regulators have enhanced the debate on the good properties of a risk measure. A fundamental issue is the evaluation of the quality of a risk estimation. On the one hand, a backtesting procedure is desirable for assessing the accuracy of such an estimation and this can be naturally achieved by elicitable risk measures. For the same objective, an alternative approach has been introduced by Davis (2016) through the so-called consistency property. On the other hand, a risk estimation should be less sensitive with respect to small changes in the available data set and exhibit qualitative robustness. A new risk measure, the Lambda value at risk (ΛV aR), has been recently proposed by Frittelli et al. (2014), as a generalization of V aR with the ability to discriminate the risk among P&L distributions with different tail behaviour. In this article, we show that ΛV aR also satisfies the properties of robustness, elicitability and consistency under some conditions.