Mathematics and Financial Economics

, Volume 9, Issue 1, pp 3–27

# Measuring risk with multiple eligible assets

• Walter Farkas
• Pablo Koch-Medina
• Cosimo Munari
Article

## Abstract

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.

## Keywords

Risk measures Multiple eligible assets Acceptance sets  Dual representations Set-valued risk measures

## Mathematics Subject Classification

91B30 46A40 46A20 46A22

## Notes

### Acknowledgments

Financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FinRisk), project “Mathematical Methods in Financial Risk Management”, is gratefully acknowledged by W. Farkas and C. Munari. Part of this research was supported by Swiss Re.

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## Authors and Affiliations

• Walter Farkas
• 1
• Pablo Koch-Medina
• 2
• Cosimo Munari
• 3
1. 1.Department of Banking and FinanceUniversity of Zurich and ETH Zurich ZurichSwitzerland
2. 2.Department of Banking and FinanceUniversity of Zurich ZurichSwitzerland
3. 3.Department of MathematicsETH Zurich ZurichSwitzerland