### Citations

2402 | Generalized Autoregressive Conditional Heteroskedasticity
- Bollerslev
- 1986
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Citation Context ...t cluster. Extensions to other light-tailed innovations ε such as GED(ν) distributions with ν > 1 are possible, cf. [10]. 2.3 The GARCH(1,1) model In the GARCH(1,1) model of Engle [12] and Bollerslev =-=[4]-=- the log price increments (Yn)n∈Z and the volatilities (σn)n∈Z are given by Yn = σnεn , σ2n = γ + αY 2 n−1 + βσ 2 n−1 , n ∈ Z , where (εn)n∈Z are iid and α, β, γ > 0. Rewriting σ2n+1 = γ + (αε 2 n + β... |

1911 |
Autoregressive Conditional Heteroscedasticity with Estimates of the variance of United Kingdom Inflation.
- Engle
- 1982
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Citation Context ...al innovations cannot cluster. Extensions to other light-tailed innovations ε such as GED(ν) distributions with ν > 1 are possible, cf. [10]. 2.3 The GARCH(1,1) model In the GARCH(1,1) model of Engle =-=[12]-=- and Bollerslev [4] the log price increments (Yn)n∈Z and the volatilities (σn)n∈Z are given by Yn = σnεn , σ2n = γ + αY 2 n−1 + βσ 2 n−1 , n ∈ Z , where (εn)n∈Z are iid and α, β, γ > 0. Rewriting σ2n+... |

1267 |
Conditional Heteroskedasticity in Asset Returns: A
- Nelson
- 1992
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Citation Context ...he same provided that the processes σ and ε are independent and η1 is light-tailed; see [9, 10, 22]. 2.2 The EGARCH model A model related to stochastic volatility models is the EGARCH model of Nelson =-=[26]-=- given by Yn = σnεn, log σ2n = α0 + ∞∑ j=1 cjg(εn−j), n ∈ Z, (8) where (εn)n∈Z is iid normal (or more generally follows a generalised error distribution GED(ν)), the real coefficients α0 and (cj)j∈N d... |

1092 |
Modelling Extremal Events For Insurance and Finance
- EMBRECHTS, KLÜPPELBERG, et al.
- 1997
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Citation Context ...tribution Φα(x) = 1(0,∞)(x) exp(−x−α) for some α > 0 (heavytailed case), a Gumbel distribution Λ(x) = exp(−e−x) (light-tailed case), or a Weibull distribution Ψα for some α > 0. We refer to the books =-=[11, 24]-=- for this and further information about extreme value theory. For θ = 1 we interpret the stochastic process as a process without clusters in the extremes, whereas if θ < 1 we speak of a process with c... |

467 | Non-Gaussian Ornstein–Uhlenbeckbasedmodels and someof their uses in financial economics.
- Barndorff-Nielsen, Shephard
- 2001
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Citation Context ...σt)t∈R, we are not aware of any explicit expressions for the tail behaviour and the extremal index function of Y1 or log Y 21 as in Theorem 1(a). 3.2 The Barndorff-Nielsen and Shephard (BNS) model In =-=[1, 2]-=- Barndorff-Nielsen and Shephard model the volatility process as a Lévy driven Ornstein-Uhlenbeck (OU) process, which results in the model Yt = ∫ (t−1,t] σs−dBs , σ2t = ∫ t −∞ e−λ(t−s)dLλs , t ∈ R , (... |

240 |
Option Pricing When the Variance Changes Randomly: Theory, Estimation, and an Application,”
- Scott
- 1987
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Citation Context ...he volatility process. Hence in the following we will often state results concerning the volatility process only. 3.1 The volatility model of Wiggins In the volatility model of Wiggins [30], see also =-=[27]-=-, the log volatility is modelled as a Gaussian Ornstein-Uhlenbeck process. More precisely, the log price increments Yt and the volatility σt are given by Yt = ∫ (t−1,t] σs− dBs, d log σ2t = (b1 − b2 l... |

212 | Econometric analysis of realised volatility and its use in estimating stochastic volatility models,
- Barndorff-Nielsen, Shephard
- 2002
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Citation Context ...σt)t∈R, we are not aware of any explicit expressions for the tail behaviour and the extremal index function of Y1 or log Y 21 as in Theorem 1(a). 3.2 The Barndorff-Nielsen and Shephard (BNS) model In =-=[1, 2]-=- Barndorff-Nielsen and Shephard model the volatility process as a Lévy driven Ornstein-Uhlenbeck (OU) process, which results in the model Yt = ∫ (t−1,t] σs−dBs , σ2t = ∫ t −∞ e−λ(t−s)dLλs , t ∈ R , (... |

104 |
Financial returns modelled by the product of two stochastic processes-a study of daily sugar prices. in:
- Taylor
- 1999
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Citation Context ...d (εn)n∈Z is iid, independent of (ηn)n∈Z. This covers the case when (log σ2n)n∈Z is a causal ARMA process with iid Gaussian noise (ηn)n∈Z, the most prominent case being the volatility model of Taylor =-=[28]-=-: Yn = σnεn, log σ2n = α0 + ψ log σ 2 n−1 + ηn, n ∈ Z, |ψ| < 1, (4) when the log volatility is a causal Gaussian AR(1) process. Extreme value analysis for (3) is based on the transformation Xn = log Y... |

93 | Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process.
- Mikosch, Starica
- 2000
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Citation Context ...E log(αε21 + β) < 0. The following result on the tail behaviour is included in Kesten’s seminal work; see [8] for further results and references. The calculation of the extremal index can be found in =-=[25, 18]-=-. 6 Claudia Klüppelberg and Alexander Lindner Theorem 3 (Tail behaviour and extremes of GARCH(1,1)). Assume the GARCH(1,1) model such that ε1 is symmetric and has a positive density on R such that E(... |

76 | Regular variation of GARCH processes.
- Basrak, Davis, et al.
- 2002
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Citation Context ... G appearing in (1) and (2) is the Fréchet distribution Φκ. We conclude that the GARCH(1,1) process is able to model clusters in the extremes. Extensions to higher order GARCH processes are given in =-=[3, 8]-=-. 3 Continuous-time models While for discrete time volatility models many results on the extremal behaviour are formulated for both the volatility process and the log price increments process, with fe... |

50 |
A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour.
- Klüppelberg, Lindner, et al.
- 2004
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Citation Context ...discrete time GARCH processes, where price and volatility are Extremes of Random Volatility Models 9 both driven by the same noise sequence (εn)n∈Z. Inspired by this, Klüppelberg, Lindner and Maller =-=[23]-=- constructed another continuous time GARCH model, termed COGARCH(1,1), which meets the features of discrete time GARCH better and for which the volatility jumps, unlike for the diffusion limit (12). L... |

30 | Extremes of stochastic volatility models
- Davis, Mikosch
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Citation Context ...sian AR(1) process. Extreme value analysis for (3) is based on the transformation Xn = log Y 2n = α0 + ∞∑ j=0 cjηn−j + log ε2n , n ∈ Z , (5) which is a Gaussian linear process plus an iid noise. From =-=[5, 7]-=- we have: Theorem 1 (Tail behaviour and extremes of the SV model). Assume the stochastic volatility model (3) as above with Xn defined by (5). (a) If ε1 is N(0, 1) denote s̃2 = s2 ∑∞ j=0 c 2 j and k =... |

25 | Extreme value theory for GARCH processes.
- Davis, Mikosch
- 2009
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Citation Context ...ome h > 0 such that the discrete skeleton (MZk (h))k∈Z has extremal index less than 1, i.e. clusters. There exist various publications on extreme value theory for time dependent data; we mention e.g. =-=[7, 8, 11, 17, 18, 19, 20, 24]-=- and references therein. 2 Discrete-time models 2.1 Stochastic volatility models The simple stochastic volatility model is given by Yn = σnεn , log σ2n = α0 + ∞∑ j=0 cjηn−j , n ∈ Z , (3) where (ηn)n∈Z... |

21 | Extremal behavior of stochastic volatility models. In:
- Fasen, Klüppelberg, et al.
- 2006
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Citation Context ...ome h > 0 such that the discrete skeleton (MZk (h))k∈Z has extremal index less than 1, i.e. clusters. There exist various publications on extreme value theory for time dependent data; we mention e.g. =-=[7, 8, 11, 17, 18, 19, 20, 24]-=- and references therein. 2 Discrete-time models 2.1 Stochastic volatility models The simple stochastic volatility model is given by Yn = σnεn , log σ2n = α0 + ∞∑ j=0 cjηn−j , n ∈ Z , (3) where (ηn)n∈Z... |

19 | Simulation methods for Levy-driven CARMA stochastic volatility models.
- Todorov, Tauchen
- 2006
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Citation Context ... occur as in Theorem 5, while for driving Lévy processes as described in Theorem 6, CARMA processes may model clusters or may not, depending on the corresponding kernel function. Todorov and Tauchen =-=[29]-=- suggest to model the volatility by a CARMA(2,1) process with a mixture of gamma distributions as driving noise process. For this model the results presented here do not apply. 3.3 Continuous-time GAR... |

14 | An exponential continuous time GARCH process. - Haug, Czado - 2007 |

11 |
Extremes of regularly varying mixed moving average processes
- Fasen
- 2005
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Citation Context ...Rs−ds+ σ20 ) e−Rt , t ≥ 0 , (13) where σ20 is independent of L. A sufficient condition for strict stationarity of (13) is the existence of some κ > 0 such that |L1|κ log+ |L1| <∞ and E(e−R1κ/2) = 1 . =-=(14)-=- Observe that the volatility in Nelson’s diffusion limit (12) has also a solution (13), with Rt defined by Rt := (ϕ+ λ2/2)t− λWt, t ≥ 0. For the stationary choice, we have E(e−R1κ/2) = 1 with κ := 2 +... |

11 | High-level dependence in time series models
- Fasen, Klüppelberg, et al.
(Show Context)
Citation Context ...ome h > 0 such that the discrete skeleton (MZk (h))k∈Z has extremal index less than 1, i.e. clusters. There exist various publications on extreme value theory for time dependent data; we mention e.g. =-=[7, 8, 11, 17, 18, 19, 20, 24]-=- and references therein. 2 Discrete-time models 2.1 Stochastic volatility models The simple stochastic volatility model is given by Yn = σnεn , log σ2n = α0 + ∞∑ j=0 cjηn−j , n ∈ Z , (3) where (ηn)n∈Z... |

10 |
Extreme Values
- Finkenstädt, Rootzén
- 2003
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9 | Asymptotic results for the sample autocovariance function and extremems of integrated generalised Ornstein-Uhlenbeck processes
- Fasen
- 2010
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Citation Context ...sion limit (12) has also a solution (13), with Rt defined by Rt := (ϕ+ λ2/2)t− λWt, t ≥ 0. For the stationary choice, we have E(e−R1κ/2) = 1 with κ := 2 + 4ϕ/λ2 > 0. (15) The following result is from =-=[16, 19]-=-: Theorem 7 (Tail and extremes of continuous time GARCH). Consider the stationary diffusion limit (12) or COGARCH(1,1) process as above with κ > 0 as given by (15) or (14), respectively. Then there ex... |

5 |
2003) \Representations of Continuous Time ARMA Processes
- Brockwell
- 2002
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Citation Context ...Then P (σ21 > x) ∼ a(x) x Eeγσ 2 1 EeγL1 P (L1 > x) , x→∞ . In particular, P (σ21 > x) = o(P (L1 > x)) for x → ∞. The extremal index function θσ is equal to 1, i.e. θσ(h) = 1 for all h > 0. Brockwell =-=[6]-=- suggests to model the volatility in (11) by Lévy driven continuous time ARMA (CARMA) processes, with the CAR(1) process being the OU process. As shown in [13, 14, 15], for CARMA processes driven by ... |

4 | Extremes of Lévy driven mixed MA processes with convolution equivalent distributions
- Fasen
(Show Context)
Citation Context ... independent of B. The volatility process is stationary and satisfies the SDE dσ2t = −λσ2t dt+ dLλt. The extremal behaviour of this model depends on the driving Lévy process and has been analysed in =-=[13, 14, 15]-=-. For regularly varying noise, as shown in [14] one obtains the following. Theorem 5 (Tail and extremes for noise in R(−α)). Consider the stationary BNS-model (11) and assume that L1 ∈ R(−α) with α > ... |

3 | Extremes of continuous-time processes
- Fasen
- 2009
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2 |
Extremwertverhalten von unendlichen Moving Average Prozessen mit leicht taillierten Innovationen und Anwendungen auf
- Drude
- 2006
(Show Context)
Citation Context ... to non-Gaussian random variables in (3) have been considered. In most cases the qualitative behaviour remains the same provided that the processes σ and ε are independent and η1 is light-tailed; see =-=[9, 10, 22]-=-. 2.2 The EGARCH model A model related to stochastic volatility models is the EGARCH model of Nelson [26] given by Yn = σnεn, log σ2n = α0 + ∞∑ j=1 cjg(εn−j), n ∈ Z, (8) where (εn)n∈Z is iid normal (o... |

2 |
Extremes of sums of infinite moving average processes with light tails and applications to EGARCH
- Drude, Lindner
- 2008
(Show Context)
Citation Context ... to non-Gaussian random variables in (3) have been considered. In most cases the qualitative behaviour remains the same provided that the processes σ and ε are independent and η1 is light-tailed; see =-=[9, 10, 22]-=-. 2.2 The EGARCH model A model related to stochastic volatility models is the EGARCH model of Nelson [26] given by Yn = σnεn, log σ2n = α0 + ∞∑ j=1 cjg(εn−j), n ∈ Z, (8) where (εn)n∈Z is iid normal (o... |

2 |
Extremes of Lévy Driven Moving Average Processes with Applications in Finance
- Fasen
- 2004
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Citation Context ...)t∈R is a continuous time process, we define for fixed h > 0 the discrete time process MZk (h) := sup (k−1)h≤t≤kh Zt , k ∈ Z . Denoting θ(h) the extremal index of the sequence (MZk (h))k∈Z, we follow =-=[13]-=- and call θ(h) for h ∈ (0,∞) the extremal index function of Z. The function θ is Extremes of Random Volatility Models 3 increasing, and we shall say that the continuous time process Z has extremal clu... |

2 |
Extreme value theory for moving avarage processes with light-tailed innovations
- Klüppelberg, Lindner
- 2005
(Show Context)
Citation Context ... to non-Gaussian random variables in (3) have been considered. In most cases the qualitative behaviour remains the same provided that the processes σ and ε are independent and η1 is light-tailed; see =-=[9, 10, 22]-=-. 2.2 The EGARCH model A model related to stochastic volatility models is the EGARCH model of Nelson [26] given by Yn = σnεn, log σ2n = α0 + ∞∑ j=1 cjg(εn−j), n ∈ Z, (8) where (εn)n∈Z is iid normal (o... |

2 |
Option values under stochastic volatility: theory and empirical estimates
- Wiggens
- 1987
(Show Context)
Citation Context ...ncentrates on the volatility process. Hence in the following we will often state results concerning the volatility process only. 3.1 The volatility model of Wiggins In the volatility model of Wiggins =-=[30]-=-, see also [27], the log volatility is modelled as a Gaussian Ornstein-Uhlenbeck process. More precisely, the log price increments Yt and the volatility σt are given by Yt = ∫ (t−1,t] σs− dBs, d log σ... |