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 exc...
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Published in | Mathematical finance Vol. 26; no. 4; pp. 867 - 900 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Oxford
Blackwell Publishing Ltd
01.10.2016
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Subjects | |
Online Access | Get full text |
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Summary: | 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 the 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, Theory Probab. Appl. 52, 614–635), Hamel and Heyde (2010, SIAM J. Financ. Math. 1, 66–95), and Hamel, Heyde, and Rudloff (2013, Math. Financ. Econ. 5, 1–28). |
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Bibliography: | Spanish Ministry of Science and Innovation - No. MTM2011-22993 and ECO2011-25706 istex:C9EAE1639C70223A1D323E8A2D987F2302B9956C Swiss National Foundation - No. 200021-137527 Chair of Excellence Programme ark:/67375/WNG-NM04J9W8-C ArticleID:MAFI12078 The authors are grateful to Leonid Hanin for his advice on duality results in Lipschitz spaces. IM acknowledges the hospitality of the University Carlos III de Madrid. IM has benefited from discussions with Qiyu Li, Michael Schmutz, and Irina Sikharulidze at various stages of this work. The second version of this preprint was greatly inspired by insightful comments of Birgit Rudloff concerning her recent work on multivariate risk measures. The authors are grateful to the Associate Editor and the referees for thoughtful comments and encouragement that led to a greatly improved paper. IC supported by the Spanish Ministry of Science and Innovation Grants No. MTM2011‐22993 and ECO2011‐25706. IM supported by the Chair of Excellence Programme of the Universidad Carlos III de Madrid and Banco Santander and the Swiss National Foundation Grant No. 200021‐137527. ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
ISSN: | 0960-1627 1467-9965 |
DOI: | 10.1111/mafi.12078 |