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Risk - A Multidisciplinary Introduction pp 133–150Cite as

The Mathematical Concept of Measuring Risk

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Abstract

One of the key tasks in risk management is the quantification of risk implied by uncertain future scenarios which then has to be interpreted with respect to certain risk management decisions. Mathematically, the usual tool for doing so is a quantitative risk measure. The financial industry standard risk measure Value-at-Risk exhibits some serious deficiencies and a vital research activity has been ongoing to search for better alternatives. In this chapter we give an introduction to the general theory of monetary, convex, and coherent risk measures and present illustrating and motivating examples.

Keywords

  • Risk measures
  • Acceptance sets
  • Robust representation
  • Risk sharing

Mathematics Subject Classification (2010)

  • 91B30
  • 91G99

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References

Selected Bibliography

  1. P. Artzner, F. Delbaen, J.M. Eber, D. Heath, Coherent measures of risks. Math. Finance 9, 203–228 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. H. Föllmer, A. Schied, Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447 (2002)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. H. Föllmer, A. Schied, Convex and coherent risk measures, in Encyclopedia of Quantitative Finance (2010), pp. 355–363

    Google Scholar 

  4. H. Föllmer, A. Schied, Stochastic Finance—An Introduction in Discrete Time, 3rd edn. (De Gruyter, Berlin, 2011)

    CrossRef  Google Scholar 

  5. M. Frittelli, E. Rosazza Giannini, Putting order in risk measures. J. Bank. Finance 26, 1473–1486 (2005)

    CrossRef  Google Scholar 

Additional Literature

  1. B. Acciaio, Optimal risk sharing with non-monotone monetary functionals. Finance Stoch. 11, 267–289 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. P. Barrieu, N. El Karoui, Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 5, 269–298 (2005)

    CrossRef  MathSciNet  Google Scholar 

  3. F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)

    CrossRef  Google Scholar 

  4. P. Carr, H. Geman, D. Madan, Pricing and hedging in incomplete markets. J. Financ. Econ. 62, 131–167 (2001)

    CrossRef  Google Scholar 

  5. V. Fasen, C. Klüppelberg, A. Menzel, Quantifying extreme risk, in Risk – A Multidisciplinary Introduction, ed. by C. Klüppelberg, D. Straub, L. Welpe (2014)

    Google Scholar 

  6. D. Filipović, G. Svindland, Optimal capital and risk allocations for law- and cash-invariant convex functions. Finance Stoch. 12, 423–439 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. D. Heath, Back to the future, in Plenary Lecture, First World Congress of the Bachelier Finance Society, Paris (2001)

    Google Scholar 

  8. E. Jouini, W. Schachermayer, N. Touzi, Optimal risk sharing for law invariant monetary utility functions. Math. Finance 18, 269–292 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. K. Larsen, T. Pirvu, S. Shreve, R. Tütüncü, Satisfying convex risk limits by trading. Finance Stoch. 9, 177–195 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. F. Knight, Risk, Uncertainty, and Profit (1921). Originally published

    Google Scholar 

  11. A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergeb. Math. (1933)

    Google Scholar 

  12. K. Mainzer, The new role of mathematical risk modeling and its importance for society, in Risk – A Multidisciplinary Introduction, ed. by C. Klüppelberg, D. Straub, L. Welpe (2014)

    Google Scholar 

  13. H.M. Markowitz, Portfolio selection. J. Finance 7, 77–91 (1952)

    Google Scholar 

  14. R.C. Merton, The theory of rational option pricing. Bell J. Econ. Manag. Sci. 7, 141–183 (1973)

    CrossRef  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Chair of Financial Mathematics, Department of Mathematics, University of München, Theresienstr 39, 80333, Munich, Germany

    Francesca Biagini

  2. Financial Mathematics, Department of Mathematics, University of München, Theresienstr 39, 80333, Munich, Germany

    Thilo Meyer-Brandis & Gregor Svindland

Authors
  1. Francesca Biagini
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  2. Thilo Meyer-Brandis
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  3. Gregor Svindland
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Corresponding author

Correspondence to Gregor Svindland .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Munich University of Technology, Munich, Germany

    Claudia Klüppelberg

  2. Faculty of Civil, Geo and Environmental Engineering, Munich University of Technology, Munich, Germany

    Daniel Straub

  3. TUM School of Management, Munich University of Technology, Munich, Germany

    Isabell M. Welpe

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Biagini, F., Meyer-Brandis, T., Svindland, G. (2014). The Mathematical Concept of Measuring Risk. In: Klüppelberg, C., Straub, D., Welpe, I. (eds) Risk - A Multidisciplinary Introduction. Springer, Cham. https://doi.org/10.1007/978-3-319-04486-6_5

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