Advances in Decision Sciences

Advances in Decision Sciences / 2006 / Article

Open Access

Volume 2006 |Article ID 086320 | https://doi.org/10.1155/JAMDS/2006/86320

A. Thavaneswaran, S. S. Appadoo, C. R. Bector, "Recent developments in volatility modeling and applications", Advances in Decision Sciences, vol. 2006, Article ID 086320, 23 pages, 2006. https://doi.org/10.1155/JAMDS/2006/86320

Recent developments in volatility modeling and applications

Received21 Feb 2006
Revised10 Jul 2006
Accepted24 Sep 2006
Published30 Nov 2006

Abstract

In financial modeling, it has been constantly pointed out that volatility clustering and conditional nonnormality induced leptokurtosis observed in high frequency data. Financial time series data are not adequately modeled by normal distribution, and empirical evidence on the non-normality assumption is well documented in the financial literature (details are illustrated by Engle (1982) and Bollerslev (1986)). An ARMA representation has been used by Thavaneswaran et al., in 2005, to derive the kurtosis of the various class of GARCH models such as power GARCH, non-Gaussian GARCH, nonstationary and random coefficient GARCH. Several empirical studies have shown that mixture distributions are more likely to capture heteroskedasticity observed in high frequency data than normal distribution. In this paper, some results on moment properties are generalized to stationary ARMA process with GARCH errors. Application to volatility forecasts and option pricing are also discussed in some detail.

References

  1. B. Abraham and A. Thavaneswaran, “A nonlinear time series model and estimation of missing observations,” Annals of the Institute of Statistical Mathematics, vol. 43, no. 3, pp. 493–504, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. S. S. Appadoo, M. Ghahramani, and A. Thavaneswaran, “Moment properties of some time series models,” The Mathematical Scientist, vol. 30, no. 1, pp. 50–63, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. S. S. Appadoo, A. Thavaneswaran, and J. Singh, “RCA models with correlated errors,” Applied Mathematics Letters, vol. 19, no. 8, pp. 824–829, 2006. View at: Publisher Site | Google Scholar
  4. T. Bollerslev, “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, vol. 31, no. 3, pp. 307–327, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. J. C. Duan, G. Gauthier, J. G. Simonato, and C. Sassevillen, “Approximate the GJR-GARCH and EGARCH option pricing models analytically,” Working Paper, University of Toronto, Ontario, 2004. View at: Google Scholar
  6. J. C. Duan and J. Wei, “Pricing foreign currency and cross-currency options under GARCH,” Journal of Derivatives, vol. 7, no. 1, pp. 51–63, 1999. View at: Google Scholar
  7. R. F. Engle, “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation,” Econometrica, vol. 50, no. 4, pp. 987–1007, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. R. F. Engle and G. Gonzalez-Rivera, “Semiparametric ARCH models,” Journal of Business and Economic Statistics, vol. 9, no. 4, pp. 345–359, 1991. View at: Publisher Site | Google Scholar
  9. P. D. Feigin and R. L. Tweedie, “Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments,” Journal of Time Series Analysis, vol. 6, no. 1, pp. 1–14, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. F. Fornari and A. Mele, “Sign- and volatility-switching ARCH models: theory and applications to international stock markets,” Journal of Applied Econometrics, vol. 12, no. 1, pp. 49–65, 1997. View at: Publisher Site | Google Scholar
  11. M. Ghahramani and A. Thavaneswaran, “Identification of ARMA models with GARCH errors,” Mathematical Scientist, vol. 32, no. 1, 2007, in preparation. View at: Google Scholar
  12. C. Gouriéroux, ARCH Models and Financial Applications, Springer Series in Statistics, Springer, New York, 1997. View at: Zentralblatt MATH | MathSciNet
  13. C. W. J. Granger, “Overview of nonlinear time series specification in economics,” in National Science Foundation Summer Symposia on Econometrics and Statistics, California, 1998. View at: Google Scholar
  14. C. W. J. Granger and T. Teräsvirta, “A simple nonlinear time series model with misleading linear properties,” Economics Letters, vol. 62, no. 2, pp. 161–165, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. C. He and T. Teräsvirta, “Properties of moments of a family of GARCH processes,” Journal of Econometrics, vol. 92, no. 1, pp. 173–192, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. S. L. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” Review of Financial Studies, vol. 6, no. 2, pp. 327–343, 1993. View at: Publisher Site | Google Scholar
  17. S. L. Heston and S. Nandi, “A closed-form GARCH option valuation model,” Review of Financial Studies, vol. 13, no. 3, pp. 585–625, 2000. View at: Publisher Site | Google Scholar
  18. R. Leipus and D. Surgailis, “Random coefficient autoregression, regime switching and long memory,” Advances in Applied Probability, vol. 35, no. 3, pp. 737–754, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. D. F. Nicholls and B. G. Quinn, Random Coefficient Autoregressive Models: An Introduction, vol. 11 of Lecture Notes in Statistics, Springer, New York, 1982. View at: Zentralblatt MATH | MathSciNet
  20. A. Thavaneswaran and B. Abraham, “Estimation for nonlinear time series models using estimating equations,” Journal of Time Series Analysis, vol. 9, no. 1, pp. 99–108, 1988. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  21. A. Thavaneswaran, S. S. Appadoo, and S. Peiris, “Forecasting volatility,” Statistics & Probability Letters, vol. 75, no. 1, pp. 1–10, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  22. A. Thavaneswaran, S. S. Appadoo, and M. Samanta, “Random coefficient GARCH models,” Mathematical and Computer Modelling, vol. 41, no. 6-7, pp. 723–733, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  23. A. Thavaneswaran and C. C. Heyde, “Prediction via estimating functions,” Journal of Statistical Planning and Inference, vol. 77, no. 1, pp. 89–101, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  24. A. Timmermann, “Moments of Markov switching models,” Journal of Econometrics, vol. 96, no. 1, pp. 75–111, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2006 A. Thavaneswaran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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