Skip to main content
SpringerLink
Log in

Set-valued average value at risk and its computation

  • Published:
Mathematics and Financial Economics Aims and scope Submit manuscript

Abstract

New versions of the set-valued average value at risk for multivariate risks are introduced by generalizing the well-known certainty equivalent representation to the set-valued case. The first ’regulator’ version is independent from any market model whereas the second version, called the market extension, takes trading opportunities into account. Essential properties of both versions are proven and an algorithmic approach is provided which admits to compute the values of both versions over finite probability spaces. Several examples illustrate various features of the theoretical constructions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. For actual computations, we used a version of the algorithm, BENSOLVE, which is available online at http://ito.mathematik.uni-halle.de/~loehne.

References

  1. Acerbi, C., Tasche, D.: On the coherence of expected shortfall. Journal of Banking and Finance 26, 1487–1503 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Global Optim. 13(1), 1–24 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burgert, C., Rüschendorf, L.: Consistent risk measures for portfolio vectors. Insurance Math. Econom. 38(2), 289–297 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ekeland, I., Galichon, A., Henry, M.: Comonotonic measures of multivariate risks. Mathematical Finance 22(1), 109–132 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ekeland, I., Schachermayer, W.: Law invariant risk measures in \(L^\infty (R_d)\). Statistics and Risk Modeling 28(3), 195–225 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Föllmer, H., Schied, A.: Stochastic Finance, 3rd edn. Walter de Gruyter & Co., Berlin (2011)

    Book  Google Scholar 

  8. Hamel, A., Heyde, F.: Duality for set-valued measures of risk. SIAM Journal on Financ. Mathematics 1, 66–95 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hamel, A., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Mathematics and Financial Economics 5, 1–28 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamel, A., Löhne, A., Rudloff, B.: A benson type algorithm for linear vector optimization and applications. Submitted for publication (2013)

  11. Hodges, S., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Futures Markets 8, 222–239 (1989)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance and Stochastics 8, 531–552 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Kabanov, Y.M.: Hedging and liquidation under transaction costs in currency markets. Finance and Stochastics 3(2), 237–248 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kabanov, Y.M., Safarian, M.: Markets with Transaction Costs: Mathematical Theory. Springer, Springer Finance (2009)

    Google Scholar 

  16. Korn, R., Müller, S.: The decoupling approach to binomial pricing of multi-asset options. J. Comput. Finance 12(3), 1–30 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Löhne, A.: Vector Optimization with Infimum and Supremum. Springer,   (2011)

  18. Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. ArXiv e-prints (2011). Submitted for publication

  19. Löhne, A., Rudloff, B.: On solvency cones and their duals in markets with transaction costs. Working paper (2013)

  20. Pflug, G.C.: Some remarks on the value-at-risk and the conditional value-at-risk. In: Probabilistic constrained optimization, Nonconvex Optim. Appl., vol. 49, pp. 272–281. Kluwer Acad. Publ., Dordrecht (2000)

  21. Rockafellar, R.T., Uryasev, S.P.: Optimization of conditional value-at-risk. Journal of risk 2, 21–42 (2000)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Yankova, M.: Set-valued Average Value at Risk with random transaction costs (2011). Senior thesis, Princeton University

Download references

Acknowledgments

The authors emphasize that this project benefited from discussions with Frank Heyde (about the primal definition of AV@R) and Andreas Löhne (about the computational part). We are grateful to the referees of a previous version for their constructive remarks. Birgit Rudloff’s research was supported by NSF award DMS-1007938.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Yeshiva University, New York, NY, USA

    Andreas H. Hamel

  2. ORFE, BCF, Princeton University, Princeton, NJ, USA

    Birgit Rudloff

  3. Barclays Capital, New York, NY, USA

    Mihaela Yankova

Authors
  1. Andreas H. Hamel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Birgit Rudloff
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mihaela Yankova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Birgit Rudloff.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hamel, A.H., Rudloff, B. & Yankova, M. Set-valued average value at risk and its computation. Math Finan Econ 7, 229–246 (2013). https://doi.org/10.1007/s11579-013-0094-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11579-013-0094-9

Keywords

Mathematics Subject Classification (2000)

JEL Classifications

Search

Navigation