Advertisement

Heavy-Tailed Distributions in VaR Calculations

  • Adam Misiorek
  • Rafał WeronEmail author
Part of the Springer Handbooks of Computational Statistics book series (SHCS)

Abstract

Market risks are the prospect of financial losses – or gains – due to unexpected changes in market prices and rates. Evaluating the exposure to such risks is nowadays of primary concern to risk managers in financial and non-financial institutions alike. Since the early 1990s a commonly used market risk estimation methodology has been the Value at Risk (VaR).

Keywords

Probability Density Function Stable Distribution Archimedean Copula Hyperbolic Distribution Generalize Inverse Gaussian 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Alexander, C.: Market Risk Analysis, Wiley, Chichester (2008)Google Scholar
  2. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk, Math. Fin. 9, 203–228 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. Atkinson, A.C.: The simulation of generalized inverse Gaussian and hyperbolic random variables. SIAM J. Sci. Stat. Comput. 3, 502–515 (1982)zbMATHCrossRefGoogle Scholar
  4. Barndorff-Nielsen, O.E.: Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond. A 353, 401–419 (1977)CrossRefGoogle Scholar
  5. Barndorff-Nielsen, O.E.: Normal ∖ ∖ Inverse Gaussian Processes and the Modelling of Stock Returns, Research Report 300, Department of Theoretical Statistics, University of Aarhus (1995)Google Scholar
  6. Barndorff-Nielsen, O.E., Blaesild, P.: Hyperbolic distributions and ramifications: Contributions to theory and applications. In: Taillie, C., Patil, G., Baldessari, B. (eds.) Statistical Distributions in Scientific Work, vol. 4, pp.19–44. Reidel, Dordrecht (1981)CrossRefGoogle Scholar
  7. Barone-Adesi, G., Giannopoulos, K., Vosper, L.: VaR without correlations for portfolios of derivative securities. J. Futures Market. 19(5), 583–602 (1999)CrossRefGoogle Scholar
  8. Basle Committee on Banking Supervision: An internal model-based approach to market risk capital requirements. http://www.bis.org. (1995)
  9. Bianchi, M.L., Rachev, S.T., Kim, Y.S., Fabozzi, F.J.: Tempered stable distributions and processes in finance: Numerical analysis. In: Corazza, M., Claudio, P.P. (eds.) Mathematical and Statistical Methods for Actuarial Sciences and Finance, Springer, New York (2010)Google Scholar
  10. Bibby, B.M., Sørensen, M.: Hyperbolic processes in finance, In: Rachev, S.T. (eds.) Handbook of Heavy-tailed Distributions in Finance, North Holland, NY, USA (2003)Google Scholar
  11. Blaesild, P., Sorensen, M.: HYP – a Computer Program for Analyzing Data by Means of the Hyperbolic Distribution, Research Report 248, Department of Theoretical Statistics, Aarhus University (1992)Google Scholar
  12. Boyarchenko, S.I., Levendorskii, S.Z.: Option pricing for truncated Lévy processes. Int. J. Theor. Appl. Fin. 3, 549–552 (2000)zbMATHCrossRefGoogle Scholar
  13. Brcich, R.F., Iskander, D.R., Zoubir, A.M.: The stability test for symmetric alpha stable distributions. IEEE Trans. Signal Process. 53, 977–986 (2005)MathSciNetCrossRefGoogle Scholar
  14. Buckle, D.J.: Bayesian inference for stable distributions. J. Am. Stat. Assoc. 90, 605–613 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: An empirical investigation, J. Bus. 75, 305–332 (2002)CrossRefGoogle Scholar
  16. Chambers, J.M., Mallows, C.L., Stuck, B.W.: A Method for Simulating Stable Random Variables. J. Am. Stat. Assoc. 71, 340–344 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  17. Chen, Y., Härdle, W., Jeong, S.-O.: Nonparametric risk management with generalized hyperbolic distributions. J. Am. Stat. Assoc. 103, 910–923 (2008)zbMATHCrossRefGoogle Scholar
  18. Christoffersen, P.: Evaluating interval forecasts. Int. Econ. Rev. 39(4), 841–862 (1998)MathSciNetCrossRefGoogle Scholar
  19. Cizek, P., Härdle, W., Weron, R.: Statistical Tools for Finance and Insurance, 2nd edition, Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
  20. Cont, R., Potters, M., Bouchaud, J.-P.: Scaling in stock market data: Stable laws and beyond, In: Dubrulle, B., Graner, F., Sornette, D. (eds.) Scale Invariance and Beyond, Proceedings of the CNRS Workshop on Scale Invariance, Springer, Berlin (1997)CrossRefGoogle Scholar
  21. D’Agostino, R.B., Stephens, M.A.: Goodness-of-Fit Techniques, Marcel Dekker, New York (1986)Google Scholar
  22. Dagpunar, J.S.: An easily implemented generalized inverse gaussian generator. Comm. Stat. Simul. 18, 703–710 (1989)MathSciNetCrossRefGoogle Scholar
  23. Danielsson, J., Hartmann, P., De Vries, C.G.: The cost of conservatism: Extreme returns, value at risk and the Basle multiplication factor. Risk 11, 101–103 (1998)Google Scholar
  24. Dominicy, Y., Veredas, D.: The Method of Simulated Quantiles, ECARES working paper, pp. 2010–008 (2010)Google Scholar
  25. DuMouchel, W.H.: Stable Distributions in Statistical Inference, Ph.D. Thesis, Department of Statistics, Yale University (1971)Google Scholar
  26. DuMouchel, W.H.: On the asymptotic normality of the maximum–likelihood estimate when sampling from a stable distribution. Ann. Stat. 1(5), 948–957 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  27. Eberlein, E., Keller, U.: Hyperbolic distributions in finance. Bernoulli 1, 281–299 (1995)zbMATHGoogle Scholar
  28. Eberlein, E., Keller, U., Prause, K.: New insights into the smile, mispricing and Value at Risk: The hyperbolic model, J. Bus. 71, 371–406 (1998)CrossRefGoogle Scholar
  29. Fama, E.F., Roll, R.: Parameter estimates for symmetric stable distributions. J. Am. Stat. Assoc. 66, 331–338 (1971)zbMATHCrossRefGoogle Scholar
  30. Fan, Z.: arameter estimation of stable distributions. Comm. Stat. Theor. Meth. 35(2), 245–255 (2006)zbMATHCrossRefGoogle Scholar
  31. Fang, K.-T., Kotz, S., Ng, K.-W.: Symmetric Multivariate and Related Distributions, Chapman & Hall, London (1987)Google Scholar
  32. Franke, J., Härdle, W., Hafner, Ch.: Statistics of Financial Markets, 2nd ed., Springer, Berlin (2008)zbMATHGoogle Scholar
  33. Fragiadakis, K., Karlis, D., Meintanis, S.G.: Tests of fit for normal inverse Gaussian distributions. Stat. Meth. 6, 553–564 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Fusai, G., Roncoroni, A.: Implementing Models in Quantitative Finance: Methods and Cases, Springer, New York (2008)zbMATHGoogle Scholar
  35. Garcia, R., Renault, E., Veredas, D.: Estimation of stable distributions by indirect inference, Journal of Econometrics 161(2), 325–337 (2011)MathSciNetCrossRefGoogle Scholar
  36. Grabchak, M.: Maximum likelihood estimation of parametric tempered stable distributions on the real line with applications to finance, Ph.D. thesis, Cornell University (2010)Google Scholar
  37. Grabchak, M., Samorodnitsky, G.: Do financial returns have finite or infinite variance? A paradox and an explanation. Quantitative Finance 10(8), 883–893 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  38. Guillaume, D.M., Dacorogna, M.M., Dave, R.R., Müller, U.A., Olsen, R.B., Pictet, O.V.: From the birds eye to the microscope: A survey of new stylized facts of the intra-daily foreign exchange markets. Fin. Stoch. 1, 95–129 (1997)zbMATHCrossRefGoogle Scholar
  39. Holt, D.R., Crow, E.L.: Tables and graphs of the stable probability density functions. J. Res. Natl. Bur. Stand. B 77B, 143–198 (1973)MathSciNetGoogle Scholar
  40. Janicki, A., Kokoszka, P.: Computer investigation of the rate of convergence of LePage type series to alpha-stable random variables. Statistica 23, 365–373 (1992)MathSciNetzbMATHGoogle Scholar
  41. Janicki, A., Weron, A.: Can one see α-stable variables and processes, Stat. Sci. 9, 109–126 (1994a)MathSciNetzbMATHCrossRefGoogle Scholar
  42. Janicki, A., Weron, A.: Simulation and Chaotic Behavior of α-Stable Stochastic Processes, Marcel Dekker (1994b)Google Scholar
  43. Joe, H.: Multivariate Models and Dependence Concepts, Chapman & Hall, London (1997)zbMATHGoogle Scholar
  44. Jorion, P.: Value at Risk: The New Benchmark for Managing Financial Risk, (3rd edn.), McGraw-Hill, NY, USA (2006)Google Scholar
  45. Karlis, D.: An EM type algorithm for maximum likelihood estimation for the Normal Inverse Gaussian distribution. Stat. Probab. Lett. 57, 43–52 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  46. Karlis, D., Lillestöl, J.: Bayesian estimation of NIG models via Markov chain Monte Carlo methods. Appl. Stoch. Models Bus. Ind. 20(4), 323–338 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  47. Kawai, R., Masuda, H.: On simulation of tempered stable random variates, Journal of Computational and Applied Mathematics 235(8), 2873-2887 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  48. Kogon, S.M., Williams, D.B.: Characteristic function based estimation of stable parameters, In: Adler, R., Feldman, R., Taqqu, M. (eds.) A Practical Guide to Heavy Tails, pp. 311–335 Birkhauser, Basel (Boston/Stuttgart) (1998)Google Scholar
  49. Koponen, I.: Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process. Phys. Rev. E 52, 1197–1199 (1995)CrossRefGoogle Scholar
  50. Koutrouvelis, I.A.: Regression–Type Estimation of the Parameters of Stable Laws J. Am. Stat. Assoc. 75, 918–928 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  51. Kuester, K., Mittnik, S., Paolella, M.S.: Value-at-Risk prediction: A comparison of alternative strategies. J. Fin. Econometrics 4(1), 53–89 (2006)CrossRefGoogle Scholar
  52. Küchler, U., Neumann, K., Sørensen, M., Streller, A.: Stock returns and hyperbolic distributions. Math. Comput. Modell. 29, 1–15 (1999)zbMATHCrossRefGoogle Scholar
  53. Lévy, P.: Calcul des Probabilites, Gauthier Villars (1925)Google Scholar
  54. Lombardi, M.J.: Bayesian inference for α-stable distributions: A random walk MCMC approach. Comput. Stat. Data Anal. 51(5), 2688–2700 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  55. Madan, D.B., Seneta, E.: The variance gamma (V.G.) model for share market returns. J. Bus. 63, 511–524 (1990)Google Scholar
  56. Mandelbrot, B.B.: The variation of certain speculative prices. J. Bus. 36, 394–419 (1963)CrossRefGoogle Scholar
  57. Mantegna, R.N.: Fast, accurate algorithm for numerical simulation of Levy stable stochastic processes. Phy. Rev. E 49, 4677–4683 (1994)CrossRefGoogle Scholar
  58. Mantegna, R.N., Stanley, H.E.: Stochastic processes with ultraslow convergence to a Gaussian: The truncated Lévy flight. Phys. Rev. Lett. 73, 2946–2949 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  59. Marshall, A.W., Olkin, I.: Families of multivariate distributions. J. Am. Stat. Assoc. 83, 834–841 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  60. Matacz, A.: Financial modeling and option theory with the truncated lévy process. Int. J. Theor. Appl. Fin. 3(1), 143–160 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  61. Matsui, M., Takemura, A.: ’Some improvements in numerical evaluation of symmetric stable density and its derivatives. Comm. Stat. Theor. Meth. 35(1), 149–172 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  62. Matsui, M., Takemura, A.: Goodness-of-fit tests for symmetric stable distributions – empirical characteristic function approach. TEST 17(3), 546–566 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  63. McCulloch, J.H.: Simple consistent estimators of stable distribution parameters. Comm. Stat. Simul. 15, 1109–1136 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  64. McNeil, A.J., Rüdiger, F., Embrechts, P.: Quantitative Risk Management, Princeton University Press, Princeton, NJ (2005)Google Scholar
  65. Michael, J.R., Schucany, W.R., Haas, R.W.: Generating random variates using transformations with multiple roots. Am. Stat. 30, 88–90 (1976)zbMATHGoogle Scholar
  66. Mittnik, S., Doganoglu, T., Chenyao, D.: Computing the Probability Density Function of the Stable Paretian Distribution. Math. Comput. Modell. 29, 235–240 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  67. Mittnik, S., Paolella, M.S.: A simple estimator for the characteristic exponent of the stable Paretian distribution. Math. Comput. Modell. 29, 161–176 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  68. Mittnik, S., Rachev, S.T., Doganoglu, T., Chenyao, D.: Maximum likelihood estimation of stable paretian models. Math. Comput. Modell. 29, 275–293 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  69. Nelsen, R.B.: An Introduction to Copulas, Springer, New York (1999)zbMATHGoogle Scholar
  70. Nolan, J.P.: Numerical calculation of stable densities and distribution functions. Comm. Stat. Stoch. Model. 13, 759–774 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  71. Nolan, J.P.: An Algorithm for Evaluating Stable Densities in Zolotarev’s (M) Parametrization. Math. Comput. Modell. 29, 229–233 (1999)zbMATHCrossRefGoogle Scholar
  72. Nolan, J.P.: Maximum Likelihood Estimation and Diagnostics for Stable Distributions. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S. (eds.) Lévy Processes, Brikhäuser, Boston (2001)Google Scholar
  73. Nolan, J.P.: Stable Distributions – Models for Heavy Tailed Data, Birkhäuser, Boston (2012); In progress, Chapter 1 online at academic2.american.edu/ ∼ jpnolan.Google Scholar
  74. Ojeda, D.: Comparison of stable estimators, Ph.D. Thesis, Department of Mathematics and Statistics, American University (2001)Google Scholar
  75. Paolella, M.S.: Testing the stable Paretian assumption. Math. Comput. Modell. 34, 1095–1112 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  76. Paolella, M.S.: Intermediate Probability: A Computational Approach, Wiley, Chichester (2007)zbMATHGoogle Scholar
  77. Peters, G.W., Sisson, S.A., Fan, Y.: Likelihood-free Bayesian inference for α-stable models, Computational Statistics & Data Analysis, forthcoming (2011)Google Scholar
  78. Poirot, J., Tankov, P.: Monte Carlo option pricing for tempered stable (CGMY) processes. Asia. Pac. Financ. Market. 13(4), 327-344 (2006)zbMATHCrossRefGoogle Scholar
  79. Prause, K.: The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures, Ph.D. Thesis, Freiburg University (1999); http://www.freidok.uni-freiburg.de/ volltexte/15.
  80. Press, S.J.: Estimation in univariate and multivariate stable distribution. J. Am. Stat. Assoc. 67, 842–846 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  81. Press, W., Teukolsky, S., Vetterling, W., Flannery, B.: Numerical Recipes in C, Cambridge University Press (1992); See also: http://www.nr.com.
  82. Pritsker, M.: The hidden dangers of historical simulation. J. Bank. Finance 30(2), 561–582 (2006)MathSciNetCrossRefGoogle Scholar
  83. Protassov, R.S.: EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ. Stat. Comput. 14, 67–77 (2004)MathSciNetCrossRefGoogle Scholar
  84. Rachev, S., Mittnik, S.: Stable Paretian Models in Finance, Wiley, New York (2000)Google Scholar
  85. Rosinski, J.: Tempering stable processes. Stoch. Process. Appl. 117(6), 677–707 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  86. Samorodnitsky, G., Taqqu, M.S.: Stable Non–Gaussian Random Processes, Chapman & Hall, London (1994)zbMATHGoogle Scholar
  87. Shuster, J.: On the inverse gaussian distribution function. J. Am. Stat. Assoc. 63, 1514–1516 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  88. Stahl, G.: Three cheers. Risk 10, 67–69 (1997)Google Scholar
  89. Stute, W., Manteiga, W.G., Quindimil, M.P.: Bootstrap based goodness-of-fit-tests. Metrika 40, 243–256 (1993)zbMATHGoogle Scholar
  90. Trivedi, P.K., Zimmer, D.M.: Copula modeling: An introduction for practitioners. Foundations Trend. Econometrics 1(1), 1–111 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  91. Venter, J.H., de Jongh, P.J.: Risk estimation using the Normal Inverse Gaussian distribution. J. Risk 4, 1–23 (2002)Google Scholar
  92. Weron, R.: On the Chambers-Mallows-Stuck Method for Simulating Skewed Stable Random Variables, Stat. Probab. Lett. 28, 165–171 (1996); See also R. Weron (1996) Correction to: On the Chambers-Mallows-Stuck Method for Simulating Skewed Stable Random Variables, Working Paper, Available at MPRA: http://mpra.ub.uni-muenchen.de/20761/.
  93. Weron, R.: Levy–Stable distributions revisited: Tail index > 2 does not exclude the Levy–Stable regime. Int. J. Modern Phys. C 12, 209–223 (2001)CrossRefGoogle Scholar
  94. Weron, R.: Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach, Wiley, Chichester (2006)Google Scholar
  95. Zolotarev, V.M.: On representation of stable laws by integrals. Selected Trans. Math. Stat. Probab. 4, 84–88 (1964)MathSciNetGoogle Scholar
  96. Zolotarev, V.M.: One–Dimensional Stable Distributions, American Mathematical Society (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Santander Consumer Bank S.A.WrocławPoland
  2. 2.Institute of Organization and ManagementWrocław University of TechnologyWrocławPoland

Personalised recommendations