# Abstracts

**Systemic Risk Measurement and Individual Capital Requirements**

*Zachary Feinstein (Princeton University, USA)*

Wednesday June 4, 14:30-15:00 | session 5.3 | Risk Measures | room EF

We define systemic risk measures for a network of interconnected banks to be the risk of the system to the obligations the financial firms have to the outside economy. Since the value that is of interest to a regulator is not the acceptable level of impact to the outside economy, but rather the capital requirements for each financial firm which makes the overall system acceptable, it is natural to consider systemic risk measures as set-valued risk measures. We will model the financial system via a network of obligations, as introduced by Eisenberg and Noe (2001), but with random endowments for each firm. In order to measure the systemic risk and create the valuations for each firm, we will introduce random stresses into the system to find the fixed-point payment structure in the network model (as a random vector). We use these clearing payments to measure the risk of the system via the resultant outcome for the outside economy. Additionally, various allocation mechanisms are discussed which can be used to define the optimal capital requirements with financial interpretations. Finally, case studies with differing network structures and stress scenarios are run to compare the acceptable capital allocations for the network of firms.

**Pareto allocations and optimal risk sharing for quasiconvex risk measures**

*Emanuela Rosazza Gianin (University of Milano-Bicocca, Italy)*

Wednesday June 4, 15:00-15:30 | session 5.3 | Risk Measures | room EF

Pareto optimal allocations and optimal risk sharing for coherent or convex risk measures as well as for insurance prices have been studied widely in the literature. In particular, Pareto optimal allocations have been characterized by applying inf-convolution of risk measures and convex analysis.

In the recent literature, an increasing interest has been devoted to quasiconvex risk measures, where convexity is replaced by quasiconvexity and cash-additivity is dropped. The main motivation to the introduction of such risk measures is that the right formulation of diversification of risk is quasiconvexity (instead of convexity) (see Cerreia-Vioglio et al. (2011), Drapeau and Kupper (2010) and Frittelli and Maggis (2011)).

The main goal of this paper is then to generalize the characterization of Pareto optimal allocations and of optimal risk sharing known for convex risk measures (see, among others, Jouini, Schachermayer and Touzi (2008)) to the quasiconvex case. Following the approach of Jouini et al. (2008) for convex risk measures, in the quasiconvex case we provide sufficient conditions for allocations to be (weakly) Pareto optimal in terms of exactness of the so-called quasiconvex inf-convolution.

We provide also some counterexamples showing that exactness of the qco-convolution does not guarantee that any allocation attaining the infimum in the qco-convolution is Pareto optimal, but only weakly Pareto optimal; and that weakly Pareto optimality does not imply in general exactness.

Moreover, we give a necessary condition for weakly optimal risk sharing that is also sufficient under cash-additivity of (at least) one between the risk measures.

**Dynamic Conic Finance via Backward Stochastic Difference Equations**

*Igor Cialenco (Illinois Institute of Technology, USA)*

Wednesday June 4, 15:30-16:00 | session 5.3 | Risk Measures | room EF

We develop a general framework, called Dynamic Conic Finance, for narrowing the theoretical spread between ask prices and bid prices of derivative securities in presence of transaction costs. We work with discrete time financial models, on a general probability space. Similar to the methods proposed in Bielecki, Cialenco, Iyigunler and Rodriguez (2013), we use the Dynamic Acceptability Indices (DAI) to define the bid and ask prices. In contrast to BCIR (2013), we consider Acceptability Indices that are sub-scale invariant, which is a desire property from practical point of view. We provide a robust representation of DAIs in terms of a family of dynamic risk measures, which consequently are linked to the solutions of a family of Backward Stochastic Difference Equations and to the notion of g-Expectation. Using these representations, we prove various properties of the proposed bid and ask prices. In particular, we show that indeed these prices do not contradict the arbitrage pricing theory, and they do shrink the super hedging pricing interval. We pay special attention to time consistency of DAI and of the corresponding dynamic risk measures. Finally, we discuss some applications of this theory, and exemplify it by using as a risk measure the entropic risk measure.