Scott, David J and Würtz, Diethelm and Dong, Christine and Tran, Thanh Tam (2009): Moments of the generalized hyperbolic distribution.

PDF
MPRA_paper_19081.pdf Download (212Kb)  Preview 
Abstract
In this paper we demonstrate a recursive method for obtaining the moments of the generalized hyperbolic distribution. The method is readily programmable for numerical evaluation of moments. For low order moments we also give an alternative derivation of the moments of the generalized hyperbolic distribution. The expressions given for these moments may be used to obtain moments for special cases such as the hyperbolic and normal inverse Gaussian distributions. Moments for limiting cases such as the skew hyperbolic t and variance gamma distributions can be found using the same approach.
Item Type:  MPRA Paper 

Original Title:  Moments of the generalized hyperbolic distribution 
Language:  English 
Keywords:  Generalized hyperbolic distribution; hyperbolic distribution; kurtosis; moments; normal inverse Gaussian distribution; skewedt distribution; skewness; Studentt distribution. 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C16  Specific Distributions 
Item ID:  19081 
Depositing User:  David J Scott 
Date Deposited:  11. Dec 2009 07:34 
Last Modified:  13. Feb 2013 07:26 
References:  Kjersti Aas and Ingrid Hobaek Haff. NIG and skew Student's t: Two special cases of the generalised hyperbolic distribution. SAMBA/01/05, January 2005. SAMBA, Norwegian Computing Center. Kjersti Aas and Ingrid Hobaek Haff. The generalised hyperbolic skew student's tdistribution. Journal of Financial Econometrics, 4(2):275{309, 2006. M. Abramowitz and I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1972. 1046 p. O. E. BarndorffNielsen. Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society of London A, 353:401{419, 1977. O. E. BarndorffNielsen. Models for nonGaussian variation; with applications to turbulence. Proc. Roy. Soc. London A, 868:501{520, 1979. O. E. BarndorffNielsen. Processes of normal inverse Gaussian type. Finance and Stochastics, 2:41{68, 1998. O. E. BarndorffNielsen and P. Blaesild. Hyperbolic distributions and ramications: Contributions to theory and applications. In C. Taillie, G. P. Patil, and B. A. Baldessari, editors, Statistical distributions in scientic work, volume 4, pages 19{44. D. Reidel, Amsterdam, 1981. O. E. BarndorffNielsen and K. Prause. Apparent scaling. Finance and Stochastics, 5:103{113, 2001. O. E. BarndorffNielsen and N. Shephard. Normal modified stable processes. Theor. Probab. Math. Statist., 65:1{19, 2001. O. E. BarndorffNielsen and R. Stelzer. Absolute moments of generalized hyperbolic distributions and approximate scaling of normal inverse Gaussian Levy processes. Scandinavian Journal of Statistics, 32(4):617{637, 2005. O. E. BarndorffNielsen, P. Blaesild, and J. Schmiegel. A parsimonious and universal description of turbulent velocity increments. European Physical Journal B, 41:345{363, 2004. B. O. Bibby and M. Sorensen. Hyperbolic processes in finance. In S. T. Rachev, editor, Handbook of Heavy Tailed Distributions in Finance, pages 211{248. Elsevier Science B. V., 2003. S. Demarta and A. J. McNeil. The tcopula and related copulas. International Statistical Review, 73(1):111{129, 2005. E. Eberlein and E. A. von Hammerstein. Generalized hyperbolic and inverse Gaussian distributions: Limiting cases and approximation of processes. FDM Preprint 80, April 2003. University of Freiburg. Ernst Eberlein and Ulrich Keller. Hyperbolic distributions in nance. Bernoulli, 1(3):281{299, Sep. 1995. M. C. Jones and M. J. Faddy. A skew extension of the t distribution, with applications. J. Royal Statist. Soc. B, 65:159{174, 2003. B. Jorgensen. Statistical properties of the generalized inverse Gaussian distribution, volume 9 of Lecture Notes in Statistics. Springer, Heidelberg, 1982. M. G. Kendall and A. Stuart. The Advanced Theory of Statistics, volume 1. Charles Griffin & Company, London, 3 edition, 1969. 439 p. Samuel Kotz, Tomasz J. Kozubowski, and Krzystof Podgorski. The Laplace Distribution and Generalizations. Birkhauser, Boston, 2001. 349 p. Dilip B. Madan and Eugene Seneta. The variance gamma (V.G.) model for share market returns. The Journal of Business, 63:511{524, 1990. A. T. McKay. A Bessel function distribution. Biometrika, 24:39{44, 1932. Alexander J. McNeil, Rudiger Frey, and Paul Embrechts. Quantitative Risk Management. Princeton Series in Finance. Princeton University Press, Princeton, NJ, 2005. 538 p. F. J. Mencia and E. Sentana. Estimation and testing of dynamic models with generalised hyperbolic innovations. CMFI Working Paper 0411, 2004. Madrid, Spain. Marc S. Paolella. Intermediate Probability: A Computational Approach. Wiley, Chichester, 2007. ISBN 9780470026373. K. Prause. The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures. PhD thesis, Universitat Freiburg, 1999. T. H. Rydberg. The normal inverse Gaussian Levy process: Simulation and approximation. Commun. Statist.Stochastic Models, 34:887{910, 1997. David J. Scott. HyperbolicDist, 2009. URL http://CRAN.Rproject.org/package= HyperbolicDist. R package version 0.62. David J. Scott and Christine Yang Dong. VarianceGamma, 2009. URL http://CRAN. Rproject.org/package=VarianceGamma. R package version 0.21. David J. Scott and Fiona Grimson. SkewHyperbolic, 2009. URL http://CRAN.Rproject.org/ package=SkewHyperbolic. R package version 0.12. Eugene Seneta. Fitting the variancegamma model to financial data. J. Appl. Probab., 41A: 177{187, 2004. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/19081 