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A note on the Estimation of a Gamma-Variance Process: Learning from a Failure

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Abstract

This paper confirms that, as originally reported in Seneta (Journal of Applied Probability 41:177–187, 2004, p. 183), it is impossible to replicate Madan et al. (European Finance Review 2:135–156, 1998) results using log daily returns on S&P 500 Index from January 1992 to September 1994. This failure leads to a close investigation of the computational problems associated with finding maximum likelihood estimates of the parameters of the popular VG model. Both standard econometric software, such as R, low level programming languages, such as Matlab\(^{\textregistered }\), and non-standard optimization software, such as Ezgrad described in Tucci (Journal of Economic Dynamics and Control 26:1739–1764, 2002), are used. The complexity of the log-likelihood function is studied. It is shown that it looks very complicated, with many local optima, and may be incredibly sensitive to very small changes in the sample used. Adding or removing a single observation may cause huge changes both in the maximum of the log-likelihood function and in the estimated parameter values. An intuitive procedure which works nicely both when implemented in R and in Matlab\(^{\textregistered }\) is presented.

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Notes

  1. See, e.g., Heyde and Liu (2001), and the references therein cited. Finlay (2009, pp. 2–3) provides a nice summary of the main points.

  2. See, e.g., Rubinstein (1994).

  3. See the appendix in Fu (2007) for a short review of the basic definitions of the Wiener, Poisson, Gamma and Lévy processes. Process of infinite activity indicates that the paths jump infinitely many times, for each finite interval and jumps that are larger than a given quantity occur only a finite number of times.

  4. See also Madan and Seneta (1990, p. 517) for some additional intuition.

  5. See Madan et al. (1998, pp. 80–81) for a summary of the main differences between Brownian motion and a VG process.

  6. See, e.g., Ribeiro and Webber (2003), Almendral and Oosterlee (2007), Kaishev and Dimitrova (2009) and the references therein cited.

  7. As correctly pointed out by an anonymous referee, the use of transformed data for financial time series may remove some relevant features present in the original untransformed data. In this case the choice is solely due to consistency with Madan et al. (1998) work.

  8. The only reason why R is used as a representative of the first group is that it is free, well documented and widely used. Matlab\(^{\textregistered }\) on the other hand is the only low level programming language, apart from good-old-fashioned FORTRAN, that the authors know.

  9. This is sometimes referred to as a subordinated GBM process. The idea of subordination was developed by Bochner (1955) and subordinated processes were first considered for stock prices by Clark (1973). See, e.g., Finlay (2009, p. 12) for more details.

  10. The VG process can be viewed as a special case of the Generalised Hyperbolic distribution (Barndorff-Nielsen and Halgreen 1977), because the Gamma distribution is a special case of the Generalised Inverse Gaussian distribution (Finlay 2009, p. 21).

  11. The parameter \( \mu \) in Eq. (6) corresponds to m in Madan et al. (1998) and c in Seneta (2004).

  12. See, e.g., Seneta (2004). These formulae are slightly different in Madan et al. (1998) where the log price increments are relative to time 0.

  13. See, e.g., Seneta (2004, p. 180).

  14. As observed in Seneta (2004, p. 180) there is some ambiguity in the terminology associated with this function. In some works it is referred to as a modified Bessel function of the second kind.

  15. See, e.g., Seneta (2004, p. 181).

  16. The parameters \(\mu \) and \(\theta \) are called m and \(\alpha \), respectively, in Madan et al. (1998). The standard deviations of parameters are omitted from Table 1. At this stage, the focus is on the estimated parameter values and associated likelihood obtained with the various methods rather than on how significantly different from zero a certain parameter is.

  17. As documented in Scott and Yang Dong (2012, pp. 16–18), it is possible to choose different sets of starting values, namely user-supplied, based on a fitted skew-Laplace distribution or derived from the method of moments, for each optimization method selected in the vgFit function. It is worth it to point out that each optimization method used, i.e. BFGS, Nelder-Mead and nlm, ends up to the same optimum regardless of the selected set of starting values.

  18. The selected optimization algorithm is Nelder–Mead in this and the following sections.

  19. Typically these are isolated cases. However in 43 cases this happens for two consecutive windows, in 6 instances for three consecutive windows and in one case for 4 consecutive windows.

  20. This is in the spirit of the algorithm that can be downloaded from http://stats.stackexchange.com/questions/30054/variance-gamma-distribution-parameter-estimation.

  21. These starting values are based on Barndorff-Nielsen and Halgreen (1977) as documented in Scott and Yang Dong (2012, p. 19).

  22. This set of parameters is used fairly often in Scott and Yang Dong (2012, p. 19).

  23. The different scales used in Figs. 7 and 9 do not allow, at this point, to compare the two best procedures.

  24. It may take from one to two seconds, for each window, to carry out these tasks depending upon the speed of the CPU.

  25. The critical value at \(99 \%\) is \(\chi _{1,99 \%}^{2}\) \(=6.635\) and at \(99.5 \%\) is \(\chi _{1,99.5 \%}^{2}\) \(=7.879.\)

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Acknowledgments

An earlier version of the present paper, entitled “Modeling log stock returns: Are parameters stable?”, was presented at the 19th International Conference on Computing in Economics and Finance held in Vancouver, CA, from July 10 through 12, 2013. The authors would like to thank the partecipants to that session for their numerous and interesting observations, some of which are reflected in the present version, and an anonymous referee for comments that greatly helped to improve the content of this paper. Moreover they would like to express their deep gratitude to Alessandro Giorgione, PhD student at the University of Sienna, for his valuable research assistance and Dott. Alberto Montesi for his highly professional librarian assistence.

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  1. Dipartimento di Economia Politica e Statistica, Università di Siena, Piazza S. Francesco, 7, 53100, Siena, Italy

    Gian P. Cervellera & Marco P. Tucci

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  1. Gian P. Cervellera
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Correspondence to Marco P. Tucci.

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Cervellera, G.P., Tucci, M.P. A note on the Estimation of a Gamma-Variance Process: Learning from a Failure. Comput Econ 49, 363–385 (2017). https://doi.org/10.1007/s10614-016-9566-3

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