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Fitting the variance-gamma model to financial data

Part of: Inference from stochastic processes Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Eugene Seneta
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Abstract

This paper has as its main theme the fitting in practice of the variance-gamma distribution, which allows for skewness, by moment methods. This fitting procedure allows for possible dependence of increments in log returns, while retaining their stationarity. It is intended as a step in a partial synthesis of some ideas of Madan, Carr and Chang (1998) and of Heyde (1999). Standard estimation and hypothesis-testing theory depends on a large sample of observations which are independently as well as identically distributed and consequently may give inappropriate conclusions in the presence of dependence.

Keywords

Variance-gammalog returnssubordinator modelskewnessincrementsdependencestationaritymethod of momentsestimationmartingalecharacteristic functiont-distribution’Student‘ processes

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions 62M05: Markov processes
Type
Part 3. Financial mathematics
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 177 - 187
Copyright
Copyright © Applied Probability Trust 2004 

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References

Carr, P. and Madan, D. B. (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 1, 61–73.CrossRefGoogle Scholar
Epps, T. W. (2000). Pricing Derivative Securities. World Scientific, Singapore.Google Scholar
Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953). Bateman Manuscript Project: Tables of Integral Transforms , 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 Gay, R. (2000). Fractals and contingent claims. Preprint.Google Scholar
Heyde, C. C. and Kou, S. G. (2002). On the controversy over tailweight of distributions. Preprint.Google Scholar
Heyde, C. C. and Leonenko, N. N. (2003). Student and related processes. Preprint.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
Hurst, S. R. (1995). The characteristic function of the Student t distribution. Res. Rep. FMRR 006-95, Centre for Financial Mathematics, The Australian National University, Canberra.Google Scholar
Hurst, S. R. and Platen, E. (1997). The marginal distribution of returns and volatility. In L1 -Statistical Procedures and Related Topics (IMS Lecture Notes Monogr. Ser. 31), ed. Dodge, Y., Institute for Mathematical Statistics, Hayward, CA, pp. 301–314.CrossRefGoogle Scholar
Hurst, S. R., Platen, E. and Rachev, S. R. (1997). Subordinated Markov models: a comparison. Financial Eng. Japanese Markets 4, 97–124.Google Scholar
Madan, D. B. and Milne, F. (1991). Option pricing with VG martingale components. Math. Finance 1, 39–55.Google Scholar
Madan, D. B. and Seneta, E. (1987a). Characteristic function estimation using maximum likelihood on transformed variables. Econometrics Discussion Papers 87-08, Department of Econometrics, University of Sydney.Google Scholar
Madan, D. B. and Seneta, E. (1987b). Chebyshev polynomial approximations and characteristic function estimation. J. R. Statist. Soc. B 49, 163–169.Google Scholar
Madan, D. B. and Seneta, E. (1987C). Simulation of estimates using the empirical characteristic function. Internat. Statist. Rev. 55, 153–161.Google Scholar
Madan, D. B. and Seneta, E. (1989). Chebyshev polynomial approximation for characteristic function estimation. Some theoretical supplements. J. R. Statist Soc. B 51, 281–285.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.Google Scholar
Nguyen, T. T., Chen, J. T., Gupta, A. K. and Dinh, K. T. (2003). A proof of the conjecture on positive skewness of generalized inverse Gaussian distributions. Biometrika 90, 245–250.Google Scholar
Praetz, P. D. (1972). The distribution of share price changes. J. Business 45, 49–55 CrossRefGoogle Scholar
Tjetjep, A. A. (2002). The variance-gamma asset pricing models. Honours Project, School of Mathematics and Statistics, University of Sydney.Google Scholar
Whittaker, E. T. and Watson, G. N. (1915). A Course of Modern Analysis. Cambridge University Press.Google Scholar