Advertisement

Probabilistic Properties of Stochastic Volatility Models

  • Richard A. DavisEmail author
  • Thomas Mikosch
Chapter

Abstract

We collect some of the probabilistic properties of a strictly stationary stochastic volatility process. These include properties about mixing, covariances and correlations, moments, and tail behavior. We also study properties of the autocovariance and autocorrelation functions of stochastic volatility processes and its powers as well as the asymptotic theory of the corresponding sample versions of these functions. In comparison with the GARCH model (see Lindner (2008)) the stochastic volatility model has a much simpler probabilistic structure which contributes to its popularity.

Keywords

Stationary Sequence Stochastic Volatility GARCH Model Stochastic Volatility Model Tail Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Basrak, B., Davis, R.A. and Mikosch, T. (2002): Regular variation of GARCH processes. Stoch. Proc. Appl. 99, 95–116.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Billingsley, P. (1968): Convergence of Probability Measures. Wiley, New York.zbMATHGoogle Scholar
  3. Billingsley, P. (1995): Probability and Measure. 3rd edition. Wiley, New York.zbMATHGoogle Scholar
  4. Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987): Regular Variation. Cambridge University Press.zbMATHGoogle Scholar
  5. Bougerol, P. and Picard, N. (1992): Stationarity of GARCH processes and of some non-negative time series. J. Econometrics 52, 115–127.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Boussama, F. (1998): Ergodicité, mélange et estimation dans le modelès GARCH. PhD Thesis, Université 7 Paris.Google Scholar
  7. Bradley, R. (2005): Basic properties of strong mixing conditions. A survey and some open questions. Probability Surveys 2, 107–144.CrossRefMathSciNetGoogle Scholar
  8. Breiman, L. (1965): On some limit theorems similar to the arc-sine law. Theory Probab. Appl. 10, 323–331.CrossRefMathSciNetGoogle Scholar
  9. Brockwell, P.J. and Davis, R.A. (1991): Time Series: Theory and Methods (2nd edition). Springer, Berlin, Heidelberg, New York.Google Scholar
  10. Davis, R.A. and Mikosch, T. (1998): Limit theory for the sample ACF of stationary process with heavy tails with applications to ARCH. Ann. Statist. 26, 2049–2080.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Davis, R.A. and Mikosch, T. (2001a): Point process convergence of stochastic volatility processes with application to sample autocorrelations. J. Appl. Probab. Special Volume: A Festschrift for David Vere-Jones 38A, 93–104.zbMATHMathSciNetGoogle Scholar
  12. Davis, R.A. and Mikosch, T. (2001b): The sample autocorrelations of financial time series models. In: Fitzgerald, W.J., Smith, R.L., Walden, A.T. and Young, P.C. (Eds.): Nonlinear and Nonstationary Signal Processing, 247–274. Cambridge University Press.Google Scholar
  13. Davis, R.A. and Mikosch, T. (2008a): Extreme value theory for GARCH models. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 186–200. Springer, New York.Google Scholar
  14. Davis, R.A. and Mikosch, T. (2008b): Extreme value theory for stochastic volatility models. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 355–364. Springer, New York.Google Scholar
  15. Doukhan, P. (1994): Mixing. Properties and Examples. Lecture Notes in Statistics 85, Springer, New York.Google Scholar
  16. Doukhan, P., Oppenheim, G. and Taqqu, M.S. (Eds.) (2004): Long Range Dependence Birkhäuser, Boston.Google Scholar
  17. Eberlein, E. and Taqqu, M.S. (Eds.) (1986): Dependence in Probability and Statistics. Birkhäuser, Boston.zbMATHGoogle Scholar
  18. Embrechts, P. and Goldie, C.M. (1980): On closure and factorization theorems for subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29, 243–256.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997): Modelling Extremal Events for Insurance and Finance. Springer, Berlin.zbMATHGoogle Scholar
  20. Fan, J. and Yao, Q. (2003): Nonlinear Time Series. Springer, Berlin, Heidelberg, New York.zbMATHCrossRefGoogle Scholar
  21. Feller, W. (1971): An Introduction to Probability Theory and Its Applications II. Wiley, New York.zbMATHGoogle Scholar
  22. Goldie, C.M. (1991): Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166.zbMATHCrossRefMathSciNetGoogle Scholar
  23. Ibragimov, I.A. and Linnik, Yu.V. (1971): Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.zbMATHGoogle Scholar
  24. Krengel, U. (1985): Ergodic Theorems. De Gruyter, Berlin.zbMATHGoogle Scholar
  25. Leland, W.E., Taqqu, M.S., Willinger, W. and Wilson, D.V. (1993): On the self-similar nature of Ethernet traffic. ACM/SIGCOMM Computer Communications Review, 183–193.Google Scholar
  26. Lindner, A.M. (2008): Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)–Processes. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 43–69. Springer, New York.Google Scholar
  27. Mandelbrot, B. (1963): The variation of certain speculative prices. J. Busin. Univ. Chicago 36, 394–419.Google Scholar
  28. Mikosch, T. (2003): Modelling dependence and tails of financial time series. In: Finkenstädt, B. and Rootzén, H. (Eds.): Extreme Values in Finance, Telecommunications and the Environment, 185–286. Chapman and Hall.Google Scholar
  29. Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002): Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. 12, 23–68.zbMATHCrossRefMathSciNetGoogle Scholar
  30. Mikosch, T. and Stărică, C. (2000): Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Stat. 28, 1427–1451.zbMATHCrossRefGoogle Scholar
  31. Mokkadem, A. (1990): Propriétés de mélange des processus autoregréssifs polynomiaux. Ann. Inst. H. Poincaré Probab. Statist. 26, 219–260.zbMATHMathSciNetGoogle Scholar
  32. Nelson, D.B. (1990): Stationarity and persistence in the GARCH(1, 1) model. Econometric Theory 6, 318–334.CrossRefMathSciNetGoogle Scholar
  33. Petrov, V.V. (1995): Limit Theorems of Probability Theory. Oxford University Press, Oxford.zbMATHGoogle Scholar
  34. Pham, T.D. and Tran, L.T. (1985): Some mixing properties of time series models. Stoch. Proc. Appl. 19, 279–303.CrossRefMathSciNetGoogle Scholar
  35. Resnick, S.I. (1987): Extreme Values, Regular Variation, and Point Processes. Springer, New York.zbMATHGoogle Scholar
  36. Samorodnitsky, G. and Taqqu, M.S. (1994): Stable Non–Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman and Hall, London.zbMATHGoogle Scholar
  37. Willinger, W., Taqqu, M.S., Sherman, R. and Wilson, D. (1995): Self-similarity through high variability: statistical analysis of ethernet lan traffic at the source level. Proceedings of the ACM/SIGCOMM‗95, Cambridge, MA. Computer Communications Review 25, 100–113.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of StatisticsColumbia UniversityNew YorkU.S.A.
  2. 2.Laboratory of Actuarial MathematicsUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations