Mathematics and Financial Economics

, Volume 9, Issue 1, pp 3–27 | Cite as

Measuring risk with multiple eligible assets

  • Walter Farkas
  • Pablo Koch-Medina
  • Cosimo Munari


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.


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

Mathematics Subject Classification

91B30 46A40 46A20 46A22 



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.


  1. 1.
    Aliprantis, C.D., Border, K.G.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)Google Scholar
  2. 2.
    Arai, T.: Good deal bounds induced by shortfall risk. SIAM J. Financ. Math. 2(1), 1–21 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Artzner, P., Delbaen, F., Koch-Medina, P.: Risk measures and efficient use of capital. ASTIN Bull. 39(1), 101–116 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Dover Publications, NewYork (2006)zbMATHGoogle Scholar
  6. 6.
    Barrieu, P., El Karoui, N.: Inf-convolution of risk measures and optimal risk transfer. Financ. Stoch. 9(2), 269–298 (2005)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bauer, H.: Sur le prolongement des formes linéaires positives dans un espace vectoriel ordonné. C. R. Acad. Sci. Paris 244, 289–292 (1957)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Clark, S.A.: The valuation problem in arbitrage price theory. J. Math. Econ. 22(5), 463–478 (1993)CrossRefzbMATHGoogle Scholar
  9. 9.
    Edgar, G.L., Sucheston, L.: Stopping Times and Directed Processes. Cambridge University Press, Cambridge (1992)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ekeland, I., Témam, R.: Convex analysis and variational problems. Society for Industrial and Applied Mathematics, Philadelphia (1999)Google Scholar
  11. 11.
    Farkas, W., Koch-Medina, P., Munari, C.: Beyond cash-additive risk measures: when changing the numéraire fails. Financ. Stoch. 18(1), 145–173 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Farkas, W., Koch-Medina, P., Munari, C.: Capital requirements with defaultable securities. Insur. Math. Econ. 55, 58–67 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Fin. Stoch. 6(4), 429–447 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. de Gruyter, NewYork (2011)CrossRefGoogle Scholar
  15. 15.
    Frittelli, M., Scandolo, G.: Risk measures and capital requirements for processes. Math. Financ. 16(4), 589–612 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1(1), 66–95 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5(1), 1–28 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hustad, O.: Linear inequalities and positive extension of linear functionals. Math. Scand. 8, 333–338 (1960)MathSciNetGoogle Scholar
  19. 19.
    Jaschke, S., Küchler, U.: Coherent risk measures and good deal bounds. Financ. Stoch. 5(2), 181–200 (2001)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kountzakis, C.E.: Generalized coherent risk measures. Appl. Math. Sci. 3(49), 2437–2451 (2009)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Kreps, D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ. 8(1), 15–35 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Namioka, I.: Partially ordered linear topological spaces, Memoirs of the American Mathematical Society, 24, Providence (1957)Google Scholar
  23. 23.
    Scandolo, G.: Models of capital requirements in static and dynamic settings. Econ. Notes 33(3), 415–435 (2004)CrossRefGoogle Scholar
  24. 24.
    Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Financ. 14(1), 19–48 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Zǎlinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

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

Personalised recommendations