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Measuring risk with multiple eligible assets

机译：使用多个合格资产衡量风险

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摘要

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.
机译：财务状况的风险是通过为满足规定的可接受性限制而筹集和投资于符合条件的交易资产组合的最低资本量来衡量的。我们针对多个合格资产调查这些风险度量的非退化性，有限性和连续性。我们的有限性和连续性结果突出了接受集和合格投资组合类别之间的相互作用。通过将封闭凸集的外部特征直接应用于接受集，我们为凸风险度量的双重表示提供了一种简单的替代方法。我们证明，当且仅当定价功能允许正扩展（这是基础接受集的支持功能）并提供此类扩展存在时的特征时，风险度量才是不变的。最后，我们讨论了用于设定值风险度量，具有不足风险的对冲以及最佳风险分担的应用。

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