Extremes of Stochastic Volatility Models
We consider extreme value theory for stochastic volatility processes in both cases of light-tailed and heavy-tailed noise. First, the asymptotic behavior of the tails of the marginal distribution is described for the two cases when the noise distribution is Gaussian or heavy-tailed. The sequence of point processes, based on the locations of the suitable normalized observations from a stochastic volatility process, converges in distribution to a Poisson process. From the point process convergence, a variety of limit results for extremes can be derived. Of special note, there is no extremal clustering for stochastic volatility processes in both the light- and heavy-tailed cases. This property is in sharp contrast with GARCH processes which exhibit extremal clustering (i.e., large values of the process come in clusters).
KeywordsPoint Process Stochastic Volatility Poisson Point Process Stochastic Volatility Model Extremal Index
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- Brockwell, P.J. and Davis, R.A. (1991): Time Series: Theory and Methods (2nd edition). Springer, Berlin, Heidelberg, New York.Google Scholar
- Davis, R.A. and Mikosch, T. (2008a): Probabilistic properties of stochastic volatility models. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 255–267. Springer, New York.Google Scholar
- Davis, R.A. and Mikosch, T. (2008b): 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
- Jensen, J.L. (1995): Saddlepoint Approximations. Oxford University Press, Oxford.Google Scholar