Stochastic volatility models with possible extremal clustering

Thomas Valentin Mikosch; Mohsen Rezapur

doi:10.3150/12-BEJ426

Stochastic volatility models with possible extremal clustering

Thomas Valentin Mikosch, Mohsen Rezapur

Department of Mathematical Sciences

12 Citations (Scopus)

Abstract

In this paper we consider a heavy-tailed stochastic volatility model, Xt = σ_tZ_t, t ε ℤ, where the volatility sequence (σ_t) and the i.i.d. noise sequence (Z_t) are assumed independent, (σ_t) is regularly varying with index α > 0, and the Z_t's have moments of order larger than α. 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 (logσ_t) is a Gaussian stationary sequence and the Zt's are regularly varying with some index α (i.e., (σ_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 σ -sequence.

Original language	English
Journal	Bernoulli
Volume	19
Issue number	5A
Pages (from-to)	1688-1713
ISSN	1350-7265
DOIs	https://doi.org/10.3150/12-BEJ426
Publication status	Published - 2013

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title = "Stochastic volatility models with possible extremal clustering",

abstract = "In this paper we consider a heavy-tailed stochastic volatility model, Xt = σtZt, t ε ℤ, where the volatility sequence (σt) and the i.i.d. noise sequence (Zt) are assumed independent, (σt) is regularly varying with index α > 0, and the Zt's have moments of order larger than α. 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 (logσt) is a Gaussian stationary sequence and the Zt's are regularly varying with some index α (i.e., (σt) has lighter tails than the Zt's), or that (Zt) is i.i.d. centered Gaussian. In these cases, we see that the sequence (Xt) does not exhibit extremal clustering. In contrast to this situation, under the conditions of this paper, both situations are possible; (Xt) may or may not have extremal clustering, depending on the clustering behavior of the σ -sequence.",

author = "Mikosch, {Thomas Valentin} and Mohsen Rezapur",

year = "2013",

doi = "10.3150/12-BEJ426",

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journal = "Bernoulli",

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TY - JOUR

T1 - Stochastic volatility models with possible extremal clustering

AU - Mikosch, Thomas Valentin

AU - Rezapur, Mohsen

PY - 2013

Y1 - 2013

N2 - In this paper we consider a heavy-tailed stochastic volatility model, Xt = σtZt, t ε ℤ, where the volatility sequence (σt) and the i.i.d. noise sequence (Zt) are assumed independent, (σt) is regularly varying with index α > 0, and the Zt's have moments of order larger than α. 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 (logσt) is a Gaussian stationary sequence and the Zt's are regularly varying with some index α (i.e., (σt) has lighter tails than the Zt's), or that (Zt) is i.i.d. centered Gaussian. In these cases, we see that the sequence (Xt) does not exhibit extremal clustering. In contrast to this situation, under the conditions of this paper, both situations are possible; (Xt) may or may not have extremal clustering, depending on the clustering behavior of the σ -sequence.

AB - In this paper we consider a heavy-tailed stochastic volatility model, Xt = σtZt, t ε ℤ, where the volatility sequence (σt) and the i.i.d. noise sequence (Zt) are assumed independent, (σt) is regularly varying with index α > 0, and the Zt's have moments of order larger than α. 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 (logσt) is a Gaussian stationary sequence and the Zt's are regularly varying with some index α (i.e., (σt) has lighter tails than the Zt's), or that (Zt) is i.i.d. centered Gaussian. In these cases, we see that the sequence (Xt) does not exhibit extremal clustering. In contrast to this situation, under the conditions of this paper, both situations are possible; (Xt) may or may not have extremal clustering, depending on the clustering behavior of the σ -sequence.

U2 - 10.3150/12-BEJ426

DO - 10.3150/12-BEJ426

M3 - Journal article

SN - 1350-7265

VL - 19

SP - 1688

EP - 1713

JO - Bernoulli

JF - Bernoulli

IS - 5A

ER -

Stochastic volatility models with possible extremal clustering

Abstract

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