Dual representations for systemic risk measures based on acceptance sets

  • Maria Arduca
  • Pablo Koch-MedinaEmail author
  • Cosimo Munari


We establish dual representations for systemic risk measures based on acceptance sets in a general setting. We deal with systemic risk measures of both “first allocate, then aggregate” and “first aggregate, then allocate” type. In both cases, we provide a detailed analysis of the corresponding systemic acceptance sets and their support functions. The same approach delivers a simple and self-contained proof of the dual representation of utility-based risk measures for univariate positions.


Systemic risk Macroprudential regulation Risk measures Dual representations 

JEL Classification

C02 G18 G32 



  1. 1.
    Aliprantis, Ch.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2006)Google Scholar
  2. 2.
    Ararat, Ç., Rudloff, B.: Dual representations for systemic risk measures. To appear in Math. Financ. Econ. arXiv:1607.03430 (2019)
  3. 3.
    Armenti, Y., Crépey, S., Drapeau, S., Papapantoleon, A.: Multivariate shortfall risk allocation and systemic risk. SIAM J. Financ. Math. 9, 90–126 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Artzner, Ph, Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Biagini, F., Fouque, J.P., Frittelli, M., Meyer-Brandis, T.: A unified approach to systemic risk measures via acceptance sets. Math. Finance 29, 329–367 (2019)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Biagini, F., Fouque, J., Frittelli, M., Meyer-Brandis, T.: On fairness of systemic risk measures. To appear in Finance Stoch. arXiv:1803.09898 (2019)
  7. 7.
    Biagini, S., Frittelli, M.: On the extension of the Namioka–Klee theorem and on the Fatou property for risk measures. In: Optimality and Risk: Modern Trends in Mathematical Finance, pp. 1–28. Springer (2009)Google Scholar
  8. 8.
    Burgert, C., Rüschendorf, L.: Consistent risk measures for portfolio vectors. Insur. Math. Econ. 38, 289–297 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, C., Iyengar, G., Moallemi, C.C.: An axiomatic approach to systemic risk. Manag. Sci. 59, 1373–1388 (2013)CrossRefGoogle Scholar
  10. 10.
    Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandmann, K., Schönbucher, P.J. (eds.) Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, pp. 1–37. Springer, Berlin (2002)Google Scholar
  11. 11.
    Delbaen, F., Owari, K.: Convex functions on dual Orlicz spaces. Positivity 23, 1051–1064 (2019)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dieudonné, J.: Sur la séparation des ensembles convexes. Math. Ann. 163(1), 1–3 (1966)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Edgar, G.A., Sucheston, L.: Stopping Times and Directed Processes. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  14. 14.
    Ekeland, I., Galichon, A., Henry, M.: Comonotonic measures of multivariate risks. Math. Finance 22, 109–132 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ekeland, I., Schachermayer, W.: Law invariant risk measures on \(L^{\infty }({\mathbb{R}}^{d})\). Stat. Risk Model. 28, 195–225 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Farkas, W., Koch-Medina, P., Munari, C.: Measuring risk with multiple eligible assets. Math. Financ. Econ. 9, 3–27 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fan, K.: Minimax theorems. Proc. Natl. Acad. Sci. USA 39, 42 (1953)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Feinstein, Z., Rudloff, B., Weber, S.: Measures of systemic risk. SIAM J. Financ. Math. 8, 672–708 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. de Gruyter, Berlin (2016)CrossRefGoogle Scholar
  20. 20.
    Frittelli, M., Scandolo, G.: Risk measures and capital requirements for processes. Math. Finance 16, 589–612 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gao, N., Leung, D., Munari, C., Xanthos, F.: Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces. Finance Stoch. 22, 395–415 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gao, N., Leung, D., Xanthos, F.: Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures. Stud. Math. 249, 329–347 (2019)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gao, N., Munari, C.: Surplus-invariant risk measures. To appear in Math. Oper. Res., arXiv:1707.04949 (2017)
  24. 24.
    Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1, 66–95 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5, 1–28 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stoch. 8, 531–552 (2004)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kromer, E., Overbeck, L., Zilch, K.: Systemic risk measures over general measurable spaces. Math. Methods Oper. Res. 84, 323–357 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Leung, D., Tantrawan, M.: On closedness of convex sets in Banach lattices. arXiv:1808.06747 (2018)
  29. 29.
    Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)CrossRefGoogle Scholar
  30. 30.
    Molchanov, I., Cascos, I.: Multivariate risk measures: a constructive approach based on selections. Math. Finance 26, 867–900 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rockafellar, R.T.: Conjugate Duality and Optimization. Society for Industrial and Applied Mathematics, Philadelphia (1974)CrossRefGoogle Scholar
  32. 32.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (2009)zbMATHGoogle Scholar
  33. 33.
    Rüschendorf, L.: Law invariant convex risk measures for portfolio vectors. Stat. Decis. 24, 97–108 (2006)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics and Quantitative MethodsUniversity of Milano-BicoccaMilanItaly
  2. 2.Center for Finance and Insurance and Swiss Finance InstituteUniversity of ZurichZurichSwitzerland

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