Multivariate risk measures: a constructive approach based on selections Articles
- MATHEMATICAL FINANCE Journal
- October 2016
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- MadridSince risky positions in multivariate portfolios can be offset by various choicesof capital requirements that depend on the exchange rules and related transactioncosts, 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 riskmeasure and so consider risk measures of set-valued portfolios. This situation includesthe classical Kabanov's transaction costs model, where the set-valued portfolio is givenby the sum of a random vector and an exchange cone, but also a number of furthercases of additional liquidity constraints. We suggest a deﬁnition of the risk measurebased on calling a set-valued portfolio acceptable if it possesses a selection with allindividually acceptable marginals. The obtained selection risk measure is coherent (orconvex), law invariant, and has values being upper convex closed sets. We describethe dual representation of the selection risk measure and suggest efﬁcient ways ofapproximating it from below and from above. In the case of Kabanov's exchange conemodel, it is shown how the selection risk measure relates to the set-valued risk measuresconsidered 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).