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Mathematical Finance
Measure of Risk
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Modeling and pricing long memory in stock market volatility
COHERENT RISK MEASURES ON GENERAL PROBABILITY SPACES
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Coherent Measures of Risk Mathematical Finance 9
Coherent Measures of Risk Mathematical Finance 9,P. Artzner,F. Delbaen,J. M. Eber,D. Heath
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Coherent Measures of Risk Mathematical Finance 9
(
Citations: 9
)
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P. Artzner
,
F. Delbaen
,
J. M. Eber
,
D. Heath
Published in 1999.
Cumulative
Annual
Citation Context
(5)
...++ − + = . (
3
) where 1,1 k θ − is the tabulated value of the corresponding elliptical percentile 1(0,1, ) Ell f and 2 () , 1/ 1/...
Fabio Lamantia
,
et al.
An Empirical Comparison among VaR Models and Time Rules with Elliptica...
...A comprehensive treatment can be found in Artzner et al. [
2
] and Delbean [9]...
...where 0 < fi < 1 (see Artzner et al. [
2
] for further details)...
Boualem Djehiche
,
et al.
Standard approaches to asset & liability risk
...Dierent authors have recently been interested in defining and constructing a coherent, in some sense, risk measure (see, for instance, Artzner et al. [
ADEH
] or F¨ollmer and Schied [FS1]), using a systematic axiomatic approach...
...where QH is the set of all additive measures such that 8 2 H, EQ ( ) 0. The risk measure v H is then coherent 2 in the sense of Artzner et al. ([
ADEH
]) and its...
Pauline Barrieu
,
et al.
Optimal Derivatives Design under Dynamic Risk Measures
...The EWMA model is an IGARCH(
1
,1) (integrated generalized autoregressive conditional heteroskedastic) model...
...The EWMA model is an IGARCH(1,
1
) (integrated generalized autoregressive conditional heteroskedastic) model...
...follows a strong IGARCH(
1
,1), that is also a particular ISRSARV(1) process (Integrated SquareRoot Stochastic Autoregressive Volatility process i.e., () 1 0...
...follows a strong IGARCH(1,
1
), that is also a particular ISRSARV(1) process (Integrated SquareRoot Stochastic Autoregressive Volatility process i.e., () 1 0...
...follows a strong IGARCH(1,1), that is also a particular ISRSARV(
1
) process (Integrated SquareRoot Stochastic Autoregressive Volatility process i.e., () 1 0...
...tk Tk Z + ∈` is still an ISRSARV(
1
) process...
...u in the high frequency IGARCH(
1
,1) process...
...u in the high frequency IGARCH(1,
1
) process...
...corresponding to the aggregated ISRSARV(
1
) process tT Z + follows the rules explained in Meddahi and Renault, 2004...
...On the other hand, the conditional variancecovariance matrix of tT Z + at time t for the EWMA model follows the rules of aggregated ISRSARV(
1
) process...
...to zero. If we assume the high frequency strong IGARCH(
1
,1) process (on the marginal distributions, or on the portfolios) 22...
...to zero. If we assume the high frequency strong IGARCH(1,
1
) process (on the marginal distributions, or on the portfolios) 22...
...In the case of strictly stationary IGARCH (
1
,1), portfolio VaR and CVaR for the temporal horizon T cannot be derived by the rules of aggregated conditional variance process explained in Meddahi and Renault, 2004...
...In the case of strictly stationary IGARCH (1,
1
), portfolio VaR and CVaR for the temporal horizon T cannot be derived by the rules of aggregated conditional variance process explained in Meddahi and Renault, 2004...
...As a matter of fact, even if the Gaussian IGARGH(
1
,1) model presents good performance at high frequency, say daily or intraday returns, the Gaussian IGARCH(1,1) is often rejected at low frequency, see Lamantia et al., 2006...
...As a matter of fact, even if the Gaussian IGARGH(1,
1
) model presents good performance at high frequency, say daily or intraday returns, the Gaussian IGARCH(1,1) is often rejected at low frequency, see Lamantia et al., 2006...
...As a matter of fact, even if the Gaussian IGARGH(1,1) model presents good performance at high frequency, say daily or intraday returns, the Gaussian IGARCH(
1
,1) is often rejected at low frequency, see Lamantia et al., 2006...
...As a matter of fact, even if the Gaussian IGARGH(1,1) model presents good performance at high frequency, say daily or intraday returns, the Gaussian IGARCH(1,
1
) is often rejected at low frequency, see Lamantia et al., 2006...
...Moreover, under these assumptions, the aggregated process is a particular ISRSARV(
1
)...
...11 (0,
1
, ); (0,1, ) Ell f Ell f � conditional value at risk values...
...11 (0,1, ); (0,
1
, ) Ell f Ell f � conditional value at risk values...
...distribution 1 (0,
1
, ) Ell f� derived from the convolution of i.i.d...
...In order to fix one for any (
1
, 2) α ∈ , we can write...
...covariance matrix. In addition, B G is also an α stable subGaussian vector where the components i ε = B i G are (
1
, 0, 0) Sα distributed, while the dispersion matrix V (that for simplicity we consider invertible) is obtained by the nn × matrix Σ i.e...
...stable subGaussian (α∈(
1
,2)) with characteristic function...
...BG ++ are (
1
, 0, 0) distributed, while the entries of dispersion matrix...
...Thus, the above model is a particular Stable GARCH(
1
,1) model (see, among others, Rachev and Mittnik, 2000) and it is also an EWMA model and an ISRSARV(1)...
...Thus, the above model is a particular Stable GARCH(1,
1
) model (see, among others, Rachev and Mittnik, 2000) and it is also an EWMA model and an ISRSARV(1)...
...Thus, the above model is a particular Stable GARCH(1,1) model (see, among others, Rachev and Mittnik, 2000) and it is also an EWMA model and an ISRSARV(
1
)...
...XS α (
1
, 0, 0) . Even in this case, we can consider the aggregated process...
...In particular, the aggregated process is still an ISRSARV(
1
) process if we consider the variance...
Fabio Lamantia
,
et al.
VaR, CVaR and Time Rules with Elliptical and Asymmetric Stable Distrib...
...Further risk measures can be found in the class of coherent measures introduced by Artzner, Delbaen, Eber and Heath [
1
] and Delbaen [5]...
Abdelali Gabih
,
et al.
Optimal portfolios with bounded Expected Loss
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Citations
(9)
An Empirical Comparison among VaR Models and Time Rules with Elliptical and Stable Distributed Returns1
(
Citations: 9
)
Fabio Lamantia
,
Sergio Ortobelli
,
Svetlozar Rachev
Published in 2006.
Standard approaches to asset & liability risk
(
Citations: 4
)
Boualem Djehiche
,
Per Hörfelt
Journal:
Scandinavian Actuarial Journal  SCAND ACTUAR J
, vol. 2005, no. 5, pp. 377400, 2005
Optimal Derivatives Design under Dynamic Risk Measures
(
Citations: 53
)
Pauline Barrieu
,
Nicole El Karoui
Published in 2004.
Draft: Coherent Risk Measures
(
Citations: 4
)
Freddy DELBAEN
Strategic LongTerm Financial Risks The OneDimensional Case
(
Citations: 5
)
Roger Kaufmann
,
Pierre Patie