Skip to main content
SpringerLink
Log in

Accounting for risk of non linear portfolios

A novel Fourier approach

  • Interdisciplinary Physics
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

The presence of non linear instruments is responsible for the emergence of non Gaussian features in the price changes distribution of realistic portfolios, even for Normally distributed risk factors. This is especially true for the benchmark Delta Gamma Normal model, which in general exhibits exponentially damped power law tails. We show how the knowledge of the model characteristic function leads to Fourier representations for two standard risk measures, the Value at Risk and the Expected Shortfall, and for their sensitivities with respect to the model parameters. We detail the numerical implementation of our formulae and we emphasize the reliability and efficiency of our results in comparison with Monte Carlo simulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, 2000)

  2. J.P. Bouchaud, M. Potters, Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management (Cambridge University Press, 2003)

  3. J. Voit, The Statistical Mechanics of Financial Markets (Springer, 2001)

  4. J. McCauley, Dynamics of Markets (Cambridge University Press, 2004)

  5. B.B. Mandelbrot, J. Business 36, 394 (1963)

    Article  Google Scholar 

  6. E.F. Fama, J. Business 38, 34 (1965)

    Article  Google Scholar 

  7. R.N. Mantegna, Physica A 179, 232 (1991)

    Article  ADS  Google Scholar 

  8. R.N. Mantegna, H.E. Stanley, Phys. Rev. Lett. 73, 2946 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. L. Borland, Phys. Rev. Lett. 89, 098701 (2002)

    Article  ADS  Google Scholar 

  10. G. Bormetti, E. Cisana, G. Montagna, O. Nicrosini, Physica A 376, 532 (2007)

    Article  ADS  Google Scholar 

  11. J.L. McCauley, G.H. Gunaratne, Physica A 329, 178 (2003)

    Article  MATH  ADS  Google Scholar 

  12. J.P. Fouque, G. Papanicolau, K.R. Sircar, Derivatives in Financial Markets with Stochastic Volatility (Cambridge University Press, 2000)

  13. A.A. Drăgulescu, V.M. Yakovenko, Quant. Finance 2, 443 (2002)

    Article  MathSciNet  Google Scholar 

  14. J. Perelló, J. Masoliver, N. Anento, Physica A 344, 134 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  15. G. Bormetti, V. Cazzola, G. Montagna, O. Nicrosini, J. Stat. Mech. P11013 (2008)

  16. J. Perelló, J. Masoliver, Phys. Rev. E 67, 037102 (2003)

    Article  ADS  Google Scholar 

  17. J.F. Muzy, J. Delour, E. Bacry, E. Phys. J. B 17, 537 (2000)

    Article  ADS  Google Scholar 

  18. B. Pochart, J.P. Bouchaud, Quant. Finance 2, 303 (2002)

    Article  MathSciNet  Google Scholar 

  19. R.N. Mantegna, H.E. Stanley, Nature 376, 46 (1995)

    Article  ADS  Google Scholar 

  20. V. Plerou, P. Gopikrishnan, M. Meyer, A. Nunes Amaral, H.E. Stanley, Phys. Rev. E 60, 5305 (1999)

    Article  ADS  Google Scholar 

  21. V. Plerou, P. Gopikrishnan, A. Nunes Amaral, M. Meyer, H.E. Stanley, Phys. Rev. E 60, 6519 (1999)

    Article  ADS  Google Scholar 

  22. P. Jorion, Value at Risk: the New Benchmark for Managing Financial Risk (McGraw-Hill, 2001)

  23. J. Mina, J.Y. Xiao, Return to RiskMetrics: the Evolution of a Standard (RiskMetrics Group, 2001)

  24. C. Acerbi, J. Banking Finance 26, 1505 (2002)

    Article  Google Scholar 

  25. S. Pafka, I. Kondor, Physica A 299, 305 (2001)

    Article  MATH  ADS  Google Scholar 

  26. A.P. Mattedi, F.M. Ramos, R.R. Rosa, R.N. Mantegna, Physica A 344, 554 (2004)

    Article  ADS  Google Scholar 

  27. G. Bormetti, V. Cazzola, G. Livan, G. Montagna, O. Nicrosini, J. Stat. Mech. P01005 (2010)

  28. G. Bormetti, M.E. De Giuli, D. Delpini, C. Tarantola, Quant. Finance, in press (2010)

  29. F.S. Lhabitant, Hedge Funds: Myths and Limits (Wiley, Chichester, 2002)

  30. J. Perelló, Physica A 383, 480 (2007)

    Article  ADS  Google Scholar 

  31. C. Rouvinez, Risk 10, 57 (1997)

    Google Scholar 

  32. R. Martin, Risk 22, 84 (2009)

    Google Scholar 

  33. S. Jaschke, J. Risk 4, 33 (2002)

    Google Scholar 

  34. S. Jaschke, C. Klüpperlberg, A. Lindner, J. Multiv. Analysis 88, 252 (2004)

    Article  MATH  Google Scholar 

  35. J. Mina, A. Ulmer, Delta-Gamma Four Ways, Working Paper RiskMetrics Group, J.P.Morgan/Reuters, available at: http://www.riskmetrics.com/research/working

  36. M. Britten-Jones, S.M. Schaefer, Eur. Finance Rev. 2, 161 (1999)

    Article  Google Scholar 

  37. A.L. Lewis, A Simple Option Formula for General Jump-Diffusion and Other Exponential Lévy Processes, (2001) available at: http://ssrn.com/abstract=282110

  38. A. Lipton, Mathematical Methods For Foreign Exchange: A Financial Engineer’s Approach (World Scientific Publishing Company, 2001)

  39. J. Major, Gradients of Risk Measures: Theory and Application to Catastrophe Risk Management and Reinsurance Pricing, Ratemaking Discussion Papers (2004)

  40. H.A. David, Order Statistics, 3rd edn. (Wiley-Interscience, 2003)

  41. P. Glasserman, P. Heidelberger, P. Shahabuddin, Math. Finance 12, 239 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  42. R.N. Mantegna, Eur. Phys. J. B 11, 193 (1999)

    Article  ADS  Google Scholar 

  43. M. Tumminello, F. Lillo, R.N. Mantegna, Phys. Rev. E 76, 031123 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  44. L. Laloux, P. Cizeau, J.P. Bouchaud, M. Potters, Phys. Rev. Lett. 83, 1467 (1998)

    Article  ADS  Google Scholar 

  45. V. Plerou, P. Gopikrishnan, B. Rosenow, A. Nunes Amaral, H.E. Stanley, Phys. Rev. Lett. 83, 1471 (1999)

    Article  ADS  Google Scholar 

  46. O. Ledoit, M. Wolf, J. Empirical Finance 10, 603 (2003)

    Article  Google Scholar 

  47. A. Meucci, Risk and Asset Allocation (Springer Finance, 2005)

Download references

Author information

Authors and Affiliations

  1. Centro Studi Rischio e Sicurezza, Istituto Universitario di Studi Superiori, V.le Lungo Ticino Sforza 56, Pavia, 27100, Italy

    G. Bormetti & V. Cazzola

  2. Istituto Nazionale di Fisica Nucleare - Sezione di Pavia, via Bassi 6, Pavia, 27100, Italy

    G. Bormetti, V. Cazzola, D. Delpini & G. Livan

  3. Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia, via Bassi 6, Pavia, 27100, Italy

    V. Cazzola, D. Delpini & G. Livan

Authors
  1. G. Bormetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Cazzola
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Delpini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Livan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Bormetti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bormetti, G., Cazzola, V., Delpini, D. et al. Accounting for risk of non linear portfolios. Eur. Phys. J. B 76, 157–165 (2010). https://doi.org/10.1140/epjb/e2010-00199-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2010-00199-9

Keywords

Search

Navigation