Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-14T16:38:13.400Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary-increment Student and variance-gamma processes

Part of:StatisticsDistribution theory - ProbabilityStochastic processesInference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Richard Finlay and
Eugene Seneta
Richard Finlay*
Affiliation:
University of Sydney
Eugene Seneta*
Affiliation:
University of Sydney
*
∗ Postal address: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia.
∗ Postal address: School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW 2006, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A continuous-time model with stationary increments for asset price {P t } is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e−rt P t } a familiar martingale. We then specify the activity time process, {T t }, for which {T t − t} is asymptotically self-similar and {τt }, with τt = T t − T t−1, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) X t and a model for {X t } which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric) t-distributions or variance-gamma distributions.

Keywords

Skew-stationary processskewnessmartingalesubordinatorvolatilitylong-range dependencet-distributionvariance-gamma distributionLaplace distributionself-similarityheavy tailminimum χ2 estimationefficient market

MSC classification

Primary:60G10: Stationary processes
Secondary:60E99: None of the above, but in this section62-07: Data analysis62M10: Time series, auto-correlation, regression, etc.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006, pp. 441 - 453
Copyright
© Applied Probability Trust 2006 

References

Barndorff-Nielsen, O. E. (1977). Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitsth. 38, 309–311.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., Kent, J. and Sørensen, M. (1982). Normal variance-mean mixtures and z distributions. Internat. Statist. Rev. 50, 145–159.Google Scholar
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.Google Scholar
Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997). The Econometrics of Financial Markets. Princeton University Press.CrossRefGoogle Scholar
Erdélyi, A., Oberhettinger, M. F. and Tricomi, F. G. (1954). Tables of Integral Transforms. Based, in part, on Notes Left by Harry Bateman and Compiled by the Staff of the Bateman Manuscript Project, Vol. 2. McGraw-Hill, New York.Google Scholar
Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 1234–1239.Google Scholar
Heyde, C. C. and Kou, S. G. (2004). On the controversy over tailweight of distributions. Operat. Res. Lett. 32, 399–408.Google Scholar
Heyde, C. C. and Leonenko, N. N. (2005). Student processes. Adv. Appl. Prob. 37, 342–365.Google Scholar
Heyde, C. C. and Liu, S. (2001). Empirical realities for a minimal description risky asset model. The need for fractal features. J. Korean Math. Soc. 38, 1047–1059.Google Scholar
Heyde, C. C. and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939–944.CrossRefGoogle Scholar
Hsiao, C. (1985). Minimum chi-square. In Encyclopedia of Statistical Sciences, Vol. 5, eds Kotz, S. and Johnson, N. L., John Wiley, New York, pp. 518–522.Google Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd edn. John Wiley, New York.Google Scholar
Madan, D. B. and Seneta, E. (1990). The variance gamma (V.G.) model for share market returns. J. Business 63, 511–524.Google Scholar
Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The variance gamma process and option pricing. Europ. Finance Rev. 2, 79–105.CrossRefGoogle Scholar
Praetz, P. D. (1972). The distribution of share price changes. J. Business 45, 49–55.Google Scholar
Schoutens, W. (2003). Lévy Processes in Finance. John Wiley, London.Google Scholar
Seneta, E. (2004). Fitting the variance-gamma model to financial data. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), eds Gani, J. and Seneta, E., Applied Probability Trust, Sheffield, pp. 177–187.Google Scholar
Sørensen, M. and Bibby, M. (2003). Hyperbolic processes in finance. In Handbook of Heavy Tailed Distributions in Finance, ed. Rachev, S. T., Elsevier, Amsterdam, pp. 211–248.Google Scholar
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.Google Scholar
Taylor, S. J. (1994). Modeling stochastic volatility: a review and comparative study. Math. Finance 4, 183–204.Google Scholar
Tjetjep, A. and Seneta, E. (2006). Skewed normal variance-mean models for asset pricing and the method of moments. Internat. Statist. Rev. 74, 109–126.Google Scholar