Extremal behavior of stochastic volatility models

  • Vicky Fasen
  • Claudia Klüppelberg
  • Alexander Lindner

Summary

Empirical volatility changes in time and exhibits tails, which are heavier than normal. Moreover, empirical volatility has — sometimes quite substantial — upwards jumps and clusters on high levels. We investigate classical and non-classical stochastic volatility models with respect to their extreme behavior. We show that classical stochastic volatility models driven by Brownian motion can model heavy tails, but obviously they are not able to model volatility jumps. Such phenomena can be modelled by Lévy driven volatility processes as, for instance, by Lévy driven Ornstein-Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving Lévy process has regularly varying tails. This results then in a volatility model with similarly heavy tails. As the last class of stochastic volatility models, we investigate a continuous time GARCH(1,1) model. Driven by an arbitrary Lévy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels.

Key words

COGARCH extreme value theory generalized Cox-Ingersoll-Ross model Lévy process Ornstein-Uhlenbeck process Poisson approximation regular variation stochastic volatility model subexponential distribution tail behavior volatility cluster 

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References

  1. [1]
    Albin, J. M. P., On extremes of infinitely divisible Ornstein-Uhlenbeck processes, Preprint, available at http://www.math.chalmers.se/~palbin/.Google Scholar
  2. [2]
    Barndorff-Nielsen, O. E., (1998), Processes of normal inverse Gaussian type, Finance Stoch., 2, 41–68.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Barndorff-Nielsen, O. E. and Shephard, N., (2001), Modelling by Lévy processes for financial econometrics. In: O. E. Bandorff-Nielsen, T. Mikosch and S. I. Resnick (Eds.), Lévy Processes: Theory and Applications, 283–318, Boston, Birkhäuser.Google Scholar
  4. [4]
    Barndorff-Nielsen, O. E. and Shephard, N., (2001), Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics (with discussion), J. Roy. Statist. Soc. Ser. B, 63(2), 167–241.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Barndorff-Nielsen, O. E. and Shephard, N., (2002), Econometric analysis of realised volatility and its use in estimating stochastic volatility models, J. Roy. Statist. Soc. Ser. B, 64, 253–280.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Bertoin, J., (1996), Lévy Processes, Cambridge University Press, Cambridge.MATHGoogle Scholar
  7. [7]
    Bingham, N. H. and Goldie, C. M. and Teugels, J. L., (1987), Regular Variation, Cambridge University Press, Cambridge.MATHGoogle Scholar
  8. [8]
    Borkovec, M. and Klüppelberg, C, (1998), Extremal behavior of diffusions models in finance, Extremes, 1(1), 47–80.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Braverman, M. and Samorodnitsky, G., (1995), Functional of infinitely divisible stochastic processes with exponential tails, Stochastic Process. Appl, 56(2), 207–231.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    Breiman, L., (1965), On some limit theorems similar to the arc-sine law, Theory Probab. Appl., 10, 323–331.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Brockwell, P. J. and Chadraa, E. and Lindner, A. M., (2005), Continuous time GARCH processes of higher order, Preprint, available at http://www.ma.tum.de/stat/.Google Scholar
  12. [12]
    Buchmann, B. and Klüppelberg, C, (2004), Fractional integral equation and state space transforms, Bernoulli, to appear.Google Scholar
  13. [13]
    Buchmann, B. and Klüppelberg, C, (2004), Maxima of stochastic processes driven by fractional Brownian motion, Adv. Appl. Probab., to appear.Google Scholar
  14. [14]
    Cline, D. B. EL, (1987), Convolution of distributions with exponential and subexponential tails, J. Austral. Math. Soc. Ser. A, 43(3), 347–365.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Cont, Ft. and Tankov, P., (2004), Financial Modelling with Jump Processes, Chapman & Hall, Boca Raton.MATHGoogle Scholar
  16. [16]
    Drost, F.C. and Werker, B. J. M., (1996), Closing the GARCH gap: continuous time GARCH modelling, J. Econometrics, 74, 31–57.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Embrechts, P. and Klüppelberg, C. and Mikosch, T., (1997), Modelling Extremal Events for Insurance and Finance, Springer, Berlin.MATHGoogle Scholar
  18. [18]
    Fasen, V., (2004), Extremes of Lévy Driven MA Processes with Applications in Finance, Ph.D. thesis, Munich University of Technology.Google Scholar
  19. [19]
    Fasen, V., (2005), Extremes of regularly varying mixed moving average processes, Preprint, available at http://www.ma.tum.de/stat/.Google Scholar
  20. [20]
    Fasen, V., (2005), Extremes of subexponential Lévy driven moving average processes, Preprint, available at http://www.ma.tum.de/stat/.Google Scholar
  21. [21]
    Goldie, C. M., (1991), Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab. 1(1), 126–166.MATHMathSciNetGoogle Scholar
  22. [22]
    Haan, L. de and Resnick, S. I. and Rootzén, H. and Vries, C. G., (1989), Extremal behavior of solutions to a stochastic difference equation with applications to ARCH processes, Stochastic Process. Appl. 32, 213–224.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Hsing, T. and Teugels, J. L., (1989), Extremal properties of shot noise processes, Adv. Appl. Probability 21, 513–525.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    Kesten, EL, (1973), Random difference equations and renewal theory for products of random matrices, Acta Math. 131, 207–248.MATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    Klüppelberg, C, (1989), Subexponential distributions and characterizations of related classes, Probab. Theory Relat. Fields 82, 259–269.MATHCrossRefGoogle Scholar
  26. [26]
    Klüppelberg, C, (2004), Risk management with extreme value theory, In: B. Finkenstädt and H. Rootzén (Eds), Extreme Values in Finance, Telecommunication and the Environment, 101–168, Chapman & Hall/CRC, Boca Raton.Google Scholar
  27. [27]
    Klüppelberg, C. and Lindner, A. and Mailer, R., (2004), A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour, J. Appl. Probab. 41(3), 601–622.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Klüppelberg, C. and Lindner, A. and Mailer, R., (2004), Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models, In: From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift (Eds. Yu. Kabanov, R. Liptser and J. Stoyanov). Springer, to appear.Google Scholar
  29. [29]
    Leadbetter, M. R. and Lindgren, G. and Rootzén, H., (1983), Extremes and Related Properties of Random Sequences and Processes, Springer, New York.MATHGoogle Scholar
  30. [30]
    Lindner, A. and Mailer, R., (2004), Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes, Preprint, available at http://www.ma.tum.de/stat/.Google Scholar
  31. [31]
    Mikosch, T. and Stărică, C., (2000), Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process, Ann. Statist. 28, 1427–1451.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    Pakes, A. G., (2004), Convolution equivalence and infinite divisibility, J. Appl. Probab. 41(2), 407–424.MATHMathSciNetCrossRefGoogle Scholar
  33. [33]
    Rootzén, H., (1986), Extreme value theory for moving average processes, Ann. Probab. 14(2), 612–652.MATHMathSciNetGoogle Scholar
  34. [34]
    Rosinski, J. and Samorodnitsky, G., (1993), Distributions of subadditive functionals of sample paths of infinitely divisible processes, Ann. Probab. 21(2), 996–1014.MathSciNetMATHGoogle Scholar
  35. [35]
    Sato, K., (1999), Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge.MATHGoogle Scholar
  36. [36]
    Vervaat, W., (1979), On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Adv. Appl. Probability 11, 750–783.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Vicky Fasen
    • 1
  • Claudia Klüppelberg
    • 1
  • Alexander Lindner
    • 1
  1. 1.Center for Mathematical SciencesMunich University of TechnologyGarchingGermany

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