Multivariate risk measures : a constructive approach based on selections

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk and so consider risk measures of set-valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. The definition of the selection risk measure is based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained risk measure is coherent (or convex), law invariant and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010), and Hamel et al. (2013)Supported by the Spanish Ministry of Science and Innovation Grants No. MTM20II—22993 and ECO20ll-25706. Supported by the Chair of Excellence Programme of the Universidad Carlos III de Madrid and Banco Santander and the Swiss National Foundation Grant No. 200021-13752

No free lunch and risk measures on Orlicz spaces

The importance of Orlicz spaces in the study of mathematics of nance came to the for in the 2000's when Frittelli and his collaborators connected the theory of utility functions to Orlicz spaces. In this thesis, we look at how Orlicz spaces play a role in nancial mathematics. After giving an overview of scalar-valued Orlicz spaces, we look at the rst fundamental theorem of asset pricing in an Orlicz space setting. We then give a brief summary of scalar risk measures, followed by the representation result for convex risk measures on Orlicz hearts. As an example of a risk measure, we take a detailed look at the Wang transform both as a pricing mechanism and as a risk measure. As the theory of nancial mathematics is moving towards the set-valued setting, we give a description of vector-valued Orlicz hearts and their duals using tensor products. Lastly, we look at set-valued risk measures on Orlicz hearts, proving a robust representation theorem via a tensor product approach