A Comparison of Techniques for Dynamic Multivariate Risk Measures

  • Zachary Feinstein
  • Birgit RudloffEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 151)


This paper contains an overview of results for dynamic multivariate risk measures. We provide the main results of four different approaches. We will prove under which assumptions results within these approaches coincide, and how properties like primal and dual representation and time consistency in the different approaches compare to each other.


Dynamic risk measures Transaction costs Set-valued risk measures Multivariate risk 

Mathematics Subject Classification (2010):

91B30 46N10 26E25 


  1. 1.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2007)zbMATHGoogle Scholar
  2. 2.
    Ararat, C., Hamel, A.H., Rudloff, B.: Set-valued shortfall and divergence risk measures (2014) (submitted for publication)Google Scholar
  3. 3.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Thinking coherently. Risk 10, 68–71 (1997)Google Scholar
  4. 4.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Artzner, P., Delbaen, F., Koch-Medina, P.: Risk measures and efficient use of capital. ASTIN Bull. 39(1), 101–116 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bálbas, A., Bálbas, R., Jiménez-Guerra, P.: Vector risk functions. Mediterr. J. Math. 9(4), 563–574 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bensaid, B., Lesne, J.-P., Pagès, H., Scheinkman, J.: Derivative asset pricing with transaction costs. Math. Financ. 2(2), 63–86 (1992)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bion-Nadal, J.: Conditional risk measures and robust representation of convex risk measures. Ecole Polytechnique, CMAP, preprint no. 557 (2004)Google Scholar
  9. 9.
    Bourbaki, N.: General Topology: Chapters 1–4. General Topology. Springer, Berlin (1998)zbMATHGoogle Scholar
  10. 10.
    Boyle, P.P., Vorst, T.: Option replication in discrete time with transaction costs. J. Financ. 47(1), 271–293 (1992)CrossRefGoogle Scholar
  11. 11.
    Burgert, C., Rüschendorf, L.: Consistent risk measures for portfolio vectors. Insur.: Math. Econ. 38(2), 289–297 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cascos, I., Molchanov, I.: Multivariate risks and depth-trimmed regions. Financ. Stoch. 11(3), 373–397 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cascos, I., Molchanov, I.: Multivariate risk measures: a constructive approach based on selections. Math. Financ. (2014), doi: 10.1111/mafi.12078
  14. 14.
    Cheridito, P., Delbaen, F., Kupper, M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11(3), 57–106 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cheridito, P., Kupper, M.: Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Financ. 14(1), 137–162 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Choquet, G.: Topology. Pure and Applied Mathematics Series. Academic Press, New York (1966)zbMATHGoogle Scholar
  17. 17.
    Detlefsen, K., Scandolo, G.: Conditional and dynamic convex risk measures. Financ. Stoch. 9(4), 539–561 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Farkas, W., Koch-Medina, P., Munari, C.: Measuring risk with multiple eligible assets. Math. Financ. Econ. 1–25 (2014)Google Scholar
  19. 19.
    Farkas, W., Koch-Medina, P., Munari, C.-A.: Beyond cash-additive risk measures: when changing the numéraire fails. Financ. Stoch. 18(1), 145–173 (2014)CrossRefzbMATHGoogle Scholar
  20. 20.
    Farkas, W., Koch-Medina, P., Munari, C.-A.: Capital requirements with defaultable securities. Insur.: Math. Econ. 55, 58–67 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Feinstein, Z., Rudloff, B.: Time consistency of dynamic risk measures in markets with transaction costs. Quant. Financ. 13(9), 1473–1489 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Feinstein, Z., Rudloff, B.: A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle (2015, submitted for publication)Google Scholar
  23. 23.
    Feinstein, Z., Rudloff, B.: Multi-portfolio time consistency for set-valued convex and coherent risk measures. Financ. Stoch. 19(1), 67–107 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Financ. Stoch. 6(4), 429–447 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Föllmer, H., Schied, A.: Stochastic Finance, 3rd edn. Walter de Gruyter & Co., Berlin (2011)CrossRefzbMATHGoogle Scholar
  26. 26.
    Frittelli, M., Scandolo, G.: Risk measures and capital requirements for processes. Math. Financ. 16(4), 589–612 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hamel, Andreas H.: A duality theory for set-valued functions I: Fenchel conjugation theory. Set-Valued Var. Anal. 17(2), 153–182 (2009)Google Scholar
  28. 28.
    Hamel, A.H., Heyde, F., Höhne, M.: Set Valued Measures of Risk. Report. Inst. für Math. (2007)Google Scholar
  29. 29.
    Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM J. Financ. Math. 1(1), 66–95 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hamel, A.H, Rudloff, B.: Continuity and finite-valuedness of set-valued risk measures. In: Tammer, C., Heyde, F. (eds.), Festschrift in Celebration of Prof. Dr. Wilfried Grecksch’s 60th Birthday, pp. 46–64. Shaker Verlag (2008)Google Scholar
  31. 31.
    Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5(1), 1–28 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hamel, A.H., Rudloff, B., Yankova, M.: Set-valued average value at risk and its computation. Math. Financ. Econ. 7(2), 229–246 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hamel, A.H., Löhne, A., Rudloff, B.: Benson type algorithms for linear vector optimization and applications. J. Glob. Optim. 59(4), 811–836 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Jacka, S., Berkaoui, A.: On representing claims for coherent risk measures, August (2007) (ArXiv e-prints)Google Scholar
  35. 35.
    Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66(1), 178–197 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Financ. Stoch. 8(4), 531–552 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kabanov, Y.M.: Hedging and liquidation under transaction costs in currency markets. Financ. Stoch. 3(2), 237–248 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kabanov, Y.M., Safarian, M.: Markets with Transaction Costs: Mathematical Theory. Springer Finance. Springer, Berlin (2009)zbMATHGoogle Scholar
  39. 39.
    Kountzakis, C.E.: Generalized coherent risk measures. Appl. Math. Sci. 3(49), 2437–2451 (2009)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Löhne, A.: Vector Optimization with Infimum and Supremum. Vector Optimization. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  41. 41.
    Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. Int. J. Theor. Appl. Financ. 17(2), 1450012 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Löhne, A., Rudloff, B., Ulus, F.: Primal and dual approximation algorithms for convex vector optimization problems. J. Glob. Optim. 60(4), 713–736 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Molchanov, I.: Theory of Random Sets. Probability and Its Applications. Springer, Berlin (2005)zbMATHGoogle Scholar
  44. 44.
    Perrakis, S., Lefoll, J.: Derivative asset pricing with transaction costs: an extension. Comput. Econ. 10(4), 359–376 (1997)CrossRefzbMATHGoogle Scholar
  45. 45.
    Riedel, F.: Dynamic coherent risk measures. Stoch. Process. Appl. 112(2), 185–200 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Roux, A.: Options Under Transaction Costs: Algorithms for Pricing and Hedging of European and American Options Under Proportional Transaction Costs and Different Borrowing and Lending Rates. VDM Verlag (2008)Google Scholar
  47. 47.
    Roux, A., Zastawniak, T.: American and Bermudan options in currency markets under proportional transaction costs, Acta Appl. Math. (2015), doi: 10.1007/s10440-015-0010-9
  48. 48.
    Ruszczynski, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31(3), 544–561 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Scandolo, G.: Models of capital requirements in static and dynamic settings. Econ. Notes 33(3), 415–435 (2004)CrossRefGoogle Scholar
  50. 50.
    Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Financ. 14(1), 19–48 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Tahar, I.B.: Tail conditional expectation for vector-valued risks. SFB 649 Discussion Papers 2006-029, Humboldt University (2006)Google Scholar
  52. 52.
    Tahar, I.B., Lépinette, E.: Vector-valued coherent risk measure processes. Int. J. Theor. Appl. Financ. 17(2), 1450011 (2014)Google Scholar
  53. 53.
    Weber, S., Anderson, W., Hamm, A.-M., Knispel, T., Liese, M., Salfeld, T.: Liquidity-adjusted risk measures. Math. Financ. Econ. 7(1), 69–91 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Yan, J.: On the commutability of essential infimum and conditional expectation operations. Chin. Sci. Bull. (Kexue Tongbao) 30(8), 1013–1018 (1985)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Electrical and Systems EngineeringWashington University in St. LouisSt. LouisUSA
  2. 2.Institute for Statistics and MathematicsVienna University of Economics and BusinessViennaAustria

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