Skip to main content

Measuring risk with multiple eligible assets

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.

This is a preview of subscription content, access via your institution.

Notes

  1. For our purposes it suffices to know that there is a linear pricing functional assigning market values to marketed payoffs. For more details on the underlying market models, also for the case of infinite dimensional marketed spaces, we refer to Clark [8] and Kreps [21].

References

  1. Aliprantis, C.D., Border, K.G.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)

    Google Scholar 

  2. Arai, T.: Good deal bounds induced by shortfall risk. SIAM J. Financ. Math. 2(1), 1–21 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Artzner, P., Delbaen, F., Koch-Medina, P.: Risk measures and efficient use of capital. ASTIN Bull. 39(1), 101–116 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Dover Publications, NewYork (2006)

    MATH  Google Scholar 

  6. Barrieu, P., El Karoui, N.: Inf-convolution of risk measures and optimal risk transfer. Financ. Stoch. 9(2), 269–298 (2005)

    Article  MATH  Google Scholar 

  7. Bauer, H.: Sur le prolongement des formes linéaires positives dans un espace vectoriel ordonné. C. R. Acad. Sci. Paris 244, 289–292 (1957)

    MATH  MathSciNet  Google Scholar 

  8. Clark, S.A.: The valuation problem in arbitrage price theory. J. Math. Econ. 22(5), 463–478 (1993)

    Article  MATH  Google Scholar 

  9. Edgar, G.L., Sucheston, L.: Stopping Times and Directed Processes. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  10. Ekeland, I., Témam, R.: Convex analysis and variational problems. Society for Industrial and Applied Mathematics, Philadelphia (1999)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Farkas, W., Koch-Medina, P., Munari, C.: Capital requirements with defaultable securities. Insur. Math. Econ. 55, 58–67 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  13. Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Fin. Stoch. 6(4), 429–447 (2002)

    Article  MATH  Google Scholar 

  14. Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. de Gruyter, NewYork (2011)

    Book  Google Scholar 

  15. Frittelli, M., Scandolo, G.: Risk measures and capital requirements for processes. Math. Financ. 16(4), 589–612 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1(1), 66–95 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5(1), 1–28 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hustad, O.: Linear inequalities and positive extension of linear functionals. Math. Scand. 8, 333–338 (1960)

    MathSciNet  Google Scholar 

  19. Jaschke, S., Küchler, U.: Coherent risk measures and good deal bounds. Financ. Stoch. 5(2), 181–200 (2001)

    Article  MATH  Google Scholar 

  20. Kountzakis, C.E.: Generalized coherent risk measures. Appl. Math. Sci. 3(49), 2437–2451 (2009)

    MATH  MathSciNet  Google Scholar 

  21. Kreps, D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ. 8(1), 15–35 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  22. Namioka, I.: Partially ordered linear topological spaces, Memoirs of the American Mathematical Society, 24, Providence (1957)

  23. Scandolo, G.: Models of capital requirements in static and dynamic settings. Econ. Notes 33(3), 415–435 (2004)

    Article  Google Scholar 

  24. Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Financ. 14(1), 19–48 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zǎlinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pablo Koch-Medina.

Additional information

Partial support through the SNF project 51NF40-144611 “Capital adequacy, valuation, and portfolio selection for insurance companies” is gratefully acknowledged.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Farkas, W., Koch-Medina, P. & Munari, C. Measuring risk with multiple eligible assets. Math Finan Econ 9, 3–27 (2015). https://doi.org/10.1007/s11579-014-0118-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11579-014-0118-0

Keywords

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

Mathematics Subject Classification

  • 91B30
  • 46A40
  • 46A20
  • 46A22