Robust market-adjusted systemic risk measures

In this note we consider a system of financial institutions and study systemic risk measures in the presence of a financial market and in a robust setting, namely, where no reference probability is assigned. We obtain a dual representation for convex robust systemic risk measures adjusted to the financial market and show its relation to some appropriate no-arbitrage conditions.

On Fairness of Systemic Risk Measures

In our previous paper, A Unified Approach to Systemic Risk Measures via Acceptance Set (textit{Mathematical Finance, 2018}), we have introduced a general class of systemic risk measures that allow for random allocations to individual banks before aggregation of their risks. In the present paper, we prove the dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximization problem which has the same optimal solution as the systemic risk measure. In addition, the optimizer in the dual formulation provides a textit{risk allocation} which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.

Disentangling Price, Risk and Model Risk: V&R measures

We propose a method to assess the intrinsic risk carried by a financial position $X$ when the agent faces uncertainty about the pricing rule assigning its present value. Our approach is inspired by a new interpretation of the quasiconvex duality in a Knightian setting, where a family of probability measures replaces the single reference probability and is then applied to value financial positions. Diametrically, our construction of Value&Risk measures is based on the selection of a basket of claims to test the reliability of models. We compare a random payoff $X$ with a given class of derivatives written on $X$ , and use these derivatives to textquotedblleft testtextquotedblright the pricing measures. We further introduce and study a general class of Value&Risk measures $% R(p,X,mathbb{P})$ that describes the additional capital that is required to make $X$ acceptable under a probability $mathbb{P}$ and given the initial price $p$ paid to acquire $X$.

Pointwise Arbitrage Pricing Theory in Discrete Time

We develop a robust framework for pricing and hedging of derivative securities in discrete-time 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 Acciao 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.

Model-free Superhedging Duality

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 $omega in Omega$, 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.

Universal Arbitrage Aggregator in Discrete Time Markets under Uncertainty

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 $mathcal{S}$ of significant sets, which we call Arbitrage de la classe $mathcal{S}$. The choice of $mathcal{S}$ reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular: for S=${Omega}$ absence of Model Independent Arbitrage is equivalent to the existence of a martingale measure; for $mathcal{S}$ 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.

Conditionally Evenly Convex Sets and Evenly Quasi-Convex Maps

Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem. This notion is then applied to obtain the dual representation of conditionally evenly quasi-convex maps.

From Risk Measures to Research Measures

In order to evaluate the quality of the scientific research, we introduce a new family of scientific performance measures, called Scientific Research Measures (SRM). Our proposal originates from the more recent developments in the theory of risk measures and is an attempt to resolve the many problems of the existing bibliometric indices. The SRM that we introduce are based on the whole scientists citation record and are: coherent, as they share the same structural properties; flexible to fit peculiarities of different areas and seniorities; granular, as they allow a more precise comparison between scientists, and inclusive, as they comprehend several popular indices. Another key feature of our SRM is that they are planned to be calibrated to the particular scientific community. We also propose a dual formulation of this problem and explain its relevance in this context.

Risk Measures on $mathcal{P}(mathbb{R})$ and Value At Risk with Probability/Loss function

We propose a generalization of the classical notion of the $V@R_{lambda}$ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The $V@R_{lambda}$ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on $mathcal{P}(% mathbb{R}).$

Complete duality for quasiconvex dynamic risk measures on modules of the $L^{p}$-type

In the conditional setting we provide a complete duality between quasiconvex risk measures defined on $L^{0}$ modules of the $L^{p}$ type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.