Lectures for Part II: Time Series Models in Finance

8.6 Extremes for GARCH and SV processes (cont)
(i) GARCH:
P(
b
(ii) SV model: P(
b
M
≤ x)
→ exp{ −γx
−1 −α
n n
M
≤ x)
→exp{
−x
−1 −α
n n
}
}
Remarks about extremal index.
(i)
(ii)
γ < 1 implies clustering of exceedances
Numerical example. Suppose c is a threshold such that
Then, if γ = .5, P(
b
(iii) 1/γ is the mean cluster size of exceedances.
(iv) Use γ to discriminate between GARCH and SV models.
(v)
P
n −1
( bn
X1
−1
n
M
n
≤ c)
~ .95
≤ c)
~ (.95)
= .975
Even for the light-tailed SV model (i.e., {Z t } ~IID N(0,1), the
extremal index is 1 (see Breidt and Davis `98 )
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