We study risk measures for financial positions in a multi-asset setting, representing the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate  finiteness and continuity properties of these multi-asset risk measures , highlighting the interplay between the acceptance set and the class of eligible portfolios. We develop a new approach to dual representations of convex multi-asset risk measures which relies on a characterization of the structure of closedconvex acceptance sets. To avoid degenerate cases we need to ensure the existence of extensions of the underlying pricing functional which belong to the effective domain of the support function of the chosen acceptance set. We provide a characterization of when such extensions exist. Finally, we discuss applications to conical market models and set-valued risk measures, optimal risk sharing, and superhedging with shortfall risk  .
The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk , and optimal risk sharing .
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We study risk measures for financial positions in a multi-asset setting, representing
The risk of financial positions is measured by
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nondegeneracy,
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multi-asset risk measures , highlighting
risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight
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develop a new approach to dual representations of convex multi-asset risk measures which relies on a characterization of the structure of closedconvex acceptance sets. To avoid degenerate cases we need to ensure the existence of extensions of the underlying pricing functional which belong to the effective domain of the support function of the chosen acceptance set. We
present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and
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conical market models and
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optimal risk sharing, and
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, and optimal risk sharing