Mathematics and Financial Economics

, Volume 8, Issue 2, pp 109–134 | Cite as

Local risk-minimization under the benchmark approach

  • Francesca BiaginiEmail author
  • Alessandra Cretarola
  • Eckhard Platen


We study the pricing and hedging of derivatives in incomplete financial markets by considering the local risk-minimization method in the context of the benchmark approach, which will be called benchmarked local risk-minimization. We show that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology.


Local risk-minimization Föllmer–Schweizer decomposition  Numéraire portfolio Benchmark approach Real world pricing 

JEL Classification

C02 G10 G13 



The authors like to thank Martin Schweizer and Wolfgang Runggaldier for valuable discussions. We also express our gratitude to an unknown referee, whose comments contributed to significant improvements of the presentation of this manuscript. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no [228087].


  1. 1.
    Ansel, J.P., Stricker, C., : Décomposition de Kunita-Watanabe. In: Séminair de Probabilités XXVII, Lecture Notes in Mathematics 1557. Springer, Berlin (1993)Google Scholar
  2. 2.
    Becherer, D.: The numeraire portfolio for unbounded semimartingales. Finance Stoch. 5(3), 327–341 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Biagini, F.: Evaluating hybrid products: the interplay between financial and insurance markets. In: Dalang, R.C., Dozzi, M., Russo, F. (eds.), Seminar on Stochastic Analysis, Random Fields and Applications VII, vol. 67 of Progress in Probability, pp. 285–304. Birkhäuser (2013)Google Scholar
  4. 4.
    Biagini, F., Cretarola, A.: Quadratic hedging methods for defaultable claims. Appl. Math. Optim. 56(3), 425–443 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Biagini, F., Pratelli, M.: Local risk minimization and numéraire. J. Appl. Probab. 36(4), 1126–1139 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Biagini, F., Widenmann, J.: Pricing of unemployment insurance products with doubly stochastic markov chains. Int. J. Theor. Appl. Finance 15(4), 1250025 (2012)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging, 1st ed. Springer Finance, Springer, Berlin, 2002 (Corr. 2nd printing) (2004)Google Scholar
  8. 8.
    Christensen, M.M., Larsen, K.: No arbitrage and the growth optimal portfolio. Stoch. Anal. Appl. 25(1), 255–280 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dellacherie, C., Meyer, P.A.: Probabilities and Potential. North Holland, Amsterdam (1978)zbMATHGoogle Scholar
  10. 10.
    Dellacherie, C., Meyer, P.A.: Probabilities and Potential B. North Holland, Amsterdam (1982)zbMATHGoogle Scholar
  11. 11.
    Du, K., Platen, E.: Benchmarked risk minimization. Math. Finance. (in press)Google Scholar
  12. 12.
    Föllmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. In: Davis, M.H.A., Elliott, R.J. (eds.) Applied Stochastic Analysis, pp. 389–414. Gordon and Breach (1991)Google Scholar
  13. 13.
    Föllmer, H., Schweizer, M.: The minimal martingale measure. Encyclopedia of Quantitative Finance, pp. 1200–1204. Wiley (2010)Google Scholar
  14. 14.
    Föllmer, H., Sondermann, D.: Hedging of non-redundant contingent claims. In: Hildenbrand, W., Mas-Colell, A. (eds.) Contributions to Mathematical Economics, pp. 205–223. North Holland (1986)Google Scholar
  15. 15.
    Goll, T., Kallsen, J.: A complete explicit solution to the log optimal portfolio problem. Ann. Appl. Probab. 13(2), 774–799 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Heath, D., Platen, E., Schweizer, M.: A comparison of two quadratic approaches to hedging in incomplete markets. Math. Finance 11(4), 385–413 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hulley, H., Schweizer, M. : \({M}^6\): on minimal market models and minimal martingale measures. In: Chiarella, C., Novikov, A. (eds.) Contemporary Quantitative Finance. Essays in Honour of Eckhard Platen, pp. 35–51. Springer (2010)Google Scholar
  18. 18.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stoch. 11(4), 447–493 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Kardaras, C.: Market viability via absence of arbitrage of the first kind. Finance Stoch. 16(4), 651–667 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Long, J.B.: The numéraire portfolio. J. Financ. Econ. 26(1), 29–69 (1990)CrossRefGoogle Scholar
  22. 22.
    Merton, R.C.: An intertemporal capital asset pricing model. Econometrica 41(5), 867–887 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Platen, E.: Diversified portfolios with jumps in a benchmark framework. Asia-Pac. Financ. Markets 11(1), 1–22 (2005)CrossRefGoogle Scholar
  24. 24.
    Platen, E.: A unifying approach to asset pricing. Research Paper Series 227, Quantitative Finance Research Centre, University of Technology, Sydney (2008)Google Scholar
  25. 25.
    Platen, E., Heath, D.: A Benchmark Approach to Quantitative Finance. Springer Finance, Berlin (2006)CrossRefzbMATHGoogle Scholar
  26. 26.
    Protter, P.: Stochastic Integration and Differential Equations. Volume 21 of Applications of Mathematics, 2nd edn. Springer, Berlin (2004)Google Scholar
  27. 27.
    Schweizer, M.: On the minimal martingale measure and the Föllmer–Schweizer decomposition. Stoch. Anal. Appl. 13, 573–599 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Schweizer, M.: A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanic, J., Musiela, M. (eds.) Option Pricing, Interest Rates and Risk Management, pp. 538–574. Cambridge University Press, Cambridge (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Francesca Biagini
    • 1
    Email author
  • Alessandra Cretarola
    • 2
  • Eckhard Platen
    • 3
  1. 1.Department of MathematicsLMUMunichGermany
  2. 2.Department of Mathematics and Computer ScienceUniversity of PerugiaPerugiaItaly
  3. 3.Finance Discipline Group, Department of Mathematical SciencesUniversity of Technology SydneyBroadwayAustralia

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