Advertisement

Optimal Bounds for Integrals with Respect to Copulas and Applications

  • Markus HoferEmail author
  • Maria Rita Iacò
Article

Abstract

We consider the integration of two-dimensional, piecewise constant functions with respect to copulas. By drawing a connection to linear assignment problems, we can give optimal upper and lower bounds for such integrals and construct the copulas for which these bounds are attained. Furthermore, we show how our approach can be extended in order to approximate extremal values in very general situations. Finally, we apply our approximation technique to problems in financial mathematics and uniform distribution theory, such as the model-independent pricing of first-to-default swaps.

Keywords

Linear assignment problems Copulas Fréchet-Hoeffding bounds Credit risk Uniform distribution theory 

Notes

Acknowledgements

The authors would like to thank Prof. Robert Tichy from TU Graz and Prof. Oto Strauch from the Slovak Academy of Science for helpful remarks and suggestions. The authors are also indebted to two anonymous referees who helped to improve the paper.

The second author is funded by the fellowship of the Doctoral School in Mathematics and Computer Science of University of Calabria and is partially supported by the Austrian Science Fund (FWF): W1230, Doctoral Program “Discrete Mathematics”.

References

  1. 1.
    Tankov, P.: Improved Fréchet bounds and model-free pricing of multi-asset options. J. Appl. Probab. 48, 389–403 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Puccetti, G., Rüschendorf, L.: Sharp bounds for sums of dependent risks. J. Appl. Probab. 50(1), 42–53 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Packham, N., Schmidt, W.M.: Latin hypercube sampling with dependence and application in finance. J. Comput. Finance 13(3), 81–111 (2010) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Rapuch, G., Roncalli, T.: Some remarks on two-asset options pricing and stochastic dependence of asset prices. Tech. report, Groupe de Recherche Operationelle, Credit Lyonnais, (2001) Google Scholar
  5. 5.
    Tchen, A.H.: Inequalities for distributions with given margins. Ann. Appl. Probab. 8, 814–827 (1980) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Fialová, J., Strauch, O.: On two-dimensional sequences composed by one-dimensional uniformly distributed sequences. Unif. Distrib. Theory 6(1), 101–125 (2011) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Albrecher, H., Asmussen, S., Kortschak, D.: Tail asymptotics for dependent subexponential differences. Sib. Math. J. 53(6), 965–983 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Puccetti, G.: Sharp bounds on the expected shortfall for a sum of dependent random variables. Stat. Probab. Lett. 83(4), 1227–1232 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Rüschendorf, L.: Solution of a statistical optimization problem by rearrangement methods. Metrika 30, 55–61 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006) zbMATHGoogle Scholar
  11. 11.
    Joe, H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London (1997) CrossRefzbMATHGoogle Scholar
  12. 12.
    Kuhn, H.W.: The Hungarian method for the assignment and transportation problems. Nav. Res. Logist. Q. 2, 83–97 (1955) CrossRefGoogle Scholar
  13. 13.
    Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia (2009) CrossRefzbMATHGoogle Scholar
  14. 14.
    Mirsky, L.: Proofs of two theorems on doubly-stochastic matrices. Proc. Am. Math. Soc. 9, 371–374 (1958) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Strauch, O., Porubský, S̆: Distribution of Sequences: A Sampler. Peter Lang, Frankfurt am Main (2005) Google Scholar
  16. 16.
    Pillichshammer, F., Steinerberger, S.: Average distance between consecutive points of uniformly distributed sequences. Unif. Distrib. Theory 4(1), 51–67 (2009) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Schmidt, W., Ward, I.: Pricing default baskets. Risk 15(1), 111–114 (2002) Google Scholar
  18. 18.
    Aistleitner, C., Hofer, M., Tichy, R.: A central limit theorem for Latin hypercube sampling with dependence and application to exotic basket option pricing. Int. J. Theor. Appl. Finance 15(7), 20 pp. (2012) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Mathematics AGraz University of TechnologyGrazAustria
  2. 2.Department of Mathematics and Computer ScienceUniversity of CalabriaArcavacata di Rende (CS)Italy

Personalised recommendations