• Bernoulli
  • Volume 19, Number 5A (2013), 1688-1713.

Stochastic volatility models with possible extremal clustering

Thomas Mikosch and Mohsen Rezapour

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In this paper we consider a heavy-tailed stochastic volatility model, $X_{t}=\sigma_{t}Z_{t}$, $t\in\mathbb{Z}$, where the volatility sequence $(\sigma_{t})$ and the i.i.d. noise sequence $(Z_{t})$ are assumed independent, $(\sigma_{t})$ is regularly varying with index $\alpha>0$, and the $Z_{t}$’s have moments of order larger than $\alpha$. In the literature (see Ann. Appl. Probab. 8 (1998) 664–675, J. Appl. Probab. 38A (2001) 93–104, In Handbook of Financial Time Series (2009) 355–364 Springer), it is typically assumed that $(\log\sigma_{t})$ is a Gaussian stationary sequence and the $Z_{t}$’s are regularly varying with some index $\alpha$ (i.e., $(\sigma_{t})$ has lighter tails than the $Z_{t}$’s), or that $(Z_{t})$ is i.i.d. centered Gaussian. In these cases, we see that the sequence $(X_{t})$ does not exhibit extremal clustering. In contrast to this situation, under the conditions of this paper, both situations are possible; $(X_{t})$ may or may not have extremal clustering, depending on the clustering behavior of the $\sigma$-sequence.

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Bernoulli, Volume 19, Number 5A (2013), 1688-1713.

First available in Project Euclid: 5 November 2013

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EGARCH exponential $\operatorname{AR}(1)$ extremal clustering extremal index $\operatorname{GARCH}$ multivariate regular variation point process stationary sequence stochastic volatility process


Mikosch, Thomas; Rezapour, Mohsen. Stochastic volatility models with possible extremal clustering. Bernoulli 19 (2013), no. 5A, 1688--1713. doi:10.3150/12-BEJ426.

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  • [1] Andersen, T.G., Davis, R.A., Kreiss, J.P. and Mikosch, T. (2009). Handbook of Financial Time Series. Berlin: Springer.
  • [2] Basrak, B., Davis, R.A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Probab. 12 908–920.
  • [3] Basrak, B., Davis, R.A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95–115.
  • [4] Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stochastic Process. Appl. 119 1055–1080.
  • [5] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [6] Boman, J. and Lindskog, F. (2009). Support theorems for the Radon transform and Cramér-Wold theorems. J. Theoret. Probab. 22 683–710.
  • [7] Breidt, F.J. and Davis, R.A. (1998). Extremes of stochastic volatility models. Ann. Appl. Probab. 8 664–675.
  • [8] Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323–331.
  • [9] Davis, R.A. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179–195.
  • [10] Davis, R.A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23 879–917.
  • [11] Davis, R.A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26 2049–2080.
  • [12] Davis, R.A. and Mikosch, T. (2001). Point process convergence of stochastic volatility processes with application to sample autocorrelation. J. Appl. Probab. 38A 93–104.
  • [13] Davis, R.A. and Mikosch, T. (2009). Extremes of stochastic volatility models. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.P. Kreiss and T. Mikosch, eds.) 355–364. Berlin: Springer.
  • [14] Davis, R.A. and Mikosch, T. (2009). Fundamental properties of stochastic volatility models. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.P. Kreiss and T. Mikosch, eds.) 255–267. Berlin: Springer.
  • [15] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. New York: Springer.
  • [16] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Berlin: Springer.
  • [17] Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55–72.
  • [18] Goldie, C.M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166.
  • [19] Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stochastic Process. Appl. 115 249–274.
  • [20] Hult, H. and Lindskog, F. (2006). On Kesten’s counterexample to the Cramér–Wold device for regular variation. Bernoulli 12 133–142.
  • [21] Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94) 121–140.
  • [22] Jessen, A.H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80(94) 171–192.
  • [23] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
  • [24] Klüppelberg, C. and Pergamenchtchikov, S. (2007). Extremal behaviour of models with multivariate random recurrence representation. Stochastic Process. Appl. 117 432–456.
  • [25] Kulik, R. and Soulier, P. (2011). The tail empirical process for long memory stochastic volatility sequences. Stochastic Process. Appl. 121 109–134.
  • [26] Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. New York: Springer.
  • [27] Mikosch, T. and Stărică, C. (2000). Limit theory for the sample autocorrelations and extremes of a $\operatorname{GARCH}(1,1)$ process. Ann. Statist. 28 1427–1451.
  • [28] Mokkadem, A. (1990). Propriétés de mélange des processus autorégressifs polynomiaux. Ann. Inst. Henri Poincaré Probab. Stat. 26 219–260.
  • [29] Nelson, D.B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347–370.
  • [30] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. New York: Springer.
  • [31] Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer.