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.
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References
Albin, J. M. P., On extremes of infinitely divisible Ornstein-Uhlenbeck processes, Preprint, available at http://www.math.chalmers.se/~palbin/.
Barndorff-Nielsen, O. E., (1998), Processes of normal inverse Gaussian type, Finance Stoch., 2, 41–68.
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.
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.
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.
Bertoin, J., (1996), Lévy Processes, Cambridge University Press, Cambridge.
Bingham, N. H. and Goldie, C. M. and Teugels, J. L., (1987), Regular Variation, Cambridge University Press, Cambridge.
Borkovec, M. and Klüppelberg, C, (1998), Extremal behavior of diffusions models in finance, Extremes, 1(1), 47–80.
Braverman, M. and Samorodnitsky, G., (1995), Functional of infinitely divisible stochastic processes with exponential tails, Stochastic Process. Appl, 56(2), 207–231.
Breiman, L., (1965), On some limit theorems similar to the arc-sine law, Theory Probab. Appl., 10, 323–331.
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/.
Buchmann, B. and Klüppelberg, C, (2004), Fractional integral equation and state space transforms, Bernoulli, to appear.
Buchmann, B. and Klüppelberg, C, (2004), Maxima of stochastic processes driven by fractional Brownian motion, Adv. Appl. Probab., to appear.
Cline, D. B. EL, (1987), Convolution of distributions with exponential and subexponential tails, J. Austral. Math. Soc. Ser. A, 43(3), 347–365.
Cont, Ft. and Tankov, P., (2004), Financial Modelling with Jump Processes, Chapman & Hall, Boca Raton.
Drost, F.C. and Werker, B. J. M., (1996), Closing the GARCH gap: continuous time GARCH modelling, J. Econometrics, 74, 31–57.
Embrechts, P. and Klüppelberg, C. and Mikosch, T., (1997), Modelling Extremal Events for Insurance and Finance, Springer, Berlin.
Fasen, V., (2004), Extremes of Lévy Driven MA Processes with Applications in Finance, Ph.D. thesis, Munich University of Technology.
Fasen, V., (2005), Extremes of regularly varying mixed moving average processes, Preprint, available at http://www.ma.tum.de/stat/.
Fasen, V., (2005), Extremes of subexponential Lévy driven moving average processes, Preprint, available at http://www.ma.tum.de/stat/.
Goldie, C. M., (1991), Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab. 1(1), 126–166.
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.
Hsing, T. and Teugels, J. L., (1989), Extremal properties of shot noise processes, Adv. Appl. Probability 21, 513–525.
Kesten, EL, (1973), Random difference equations and renewal theory for products of random matrices, Acta Math. 131, 207–248.
Klüppelberg, C, (1989), Subexponential distributions and characterizations of related classes, Probab. Theory Relat. Fields 82, 259–269.
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.
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.
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.
Leadbetter, M. R. and Lindgren, G. and Rootzén, H., (1983), Extremes and Related Properties of Random Sequences and Processes, Springer, New York.
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/.
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.
Pakes, A. G., (2004), Convolution equivalence and infinite divisibility, J. Appl. Probab. 41(2), 407–424.
Rootzén, H., (1986), Extreme value theory for moving average processes, Ann. Probab. 14(2), 612–652.
Rosinski, J. and Samorodnitsky, G., (1993), Distributions of subadditive functionals of sample paths of infinitely divisible processes, Ann. Probab. 21(2), 996–1014.
Sato, K., (1999), Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge.
Vervaat, W., (1979), On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Adv. Appl. Probability 11, 750–783.
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Fasen, V., Klüppelberg, C., Lindner, A. (2006). Extremal behavior of stochastic volatility models. In: Shiryaev, A.N., Grossinho, M.R., Oliveira, P.E., Esquível, M.L. (eds) Stochastic Finance. Springer, Boston, MA. https://doi.org/10.1007/0-387-28359-5_4
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DOI: https://doi.org/10.1007/0-387-28359-5_4
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