2013/01/08 by Ignacio Cascos, Ilya Molchanov, Cascos, Ignacio +1 · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60D05 #60E99 #91B30 #91G99 #FOS: Economics and business #FOS: Mathematics #Fuzzy Systems and Optimization #Multi-Criteria Decision Making #Probability (math.PR) #Risk Management (q-fin.RM) #Risk and Portfolio Optimization #math.PR #msc:60D05 #msc:60E99 #msc:91B30 #msc:91G99 #q-fin.RM
paper · pdf · doi:10.48550/arxiv.1301.1496
38 pages, 5 figures. Corrections to Section 7 concerning the duality results
openalex publication_date 2013/01/08 · arxiv created 2016/07/11 · arxiv updated 2016/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 measure 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. We suggest a definition of the risk measure based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection 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, Heyde and Rudloff (2013).