By D. Sornette, Y. Malevergne
"Clearly elucidates severe monetary dangers linked to infrequent occasions similar to monetary crashes. The spotlight of the e-book is the delineation of assorted copulas along with monetary dependences between various resources of a portfolio. specifically, the insightful dialogue on quadrant and orthant dependences casts new mild at the connection among marginal types and monetary dependence...brings a brilliant portrayal of the subject." -- MATHEMATICAL stories
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Additional resources for Extreme Financial Risks: From Dependence to Risk Management (Springer Finance)
5. This relationship can be derived from the assumption that agents are proﬁt maximizers. These two ingredients FV (V ) ∼ 1 − 1/V γ and r = kV β imply that large price returns r have also a power law distribution with exponent µ = γ/β ≈ 3. Gabaix et al. ﬁnd that their theory is consistent with the data. It is important to stress that these results are obtained by using aggregated data over a ﬁxed time interval. Farmer and Lillo  argue that aggregating the data in time complicates the discussion, since the functional form of the market impact generally depends on the length of the time interval.
However, for real ﬁnancial data, the convergence in distribution to a Gaussian law is very slow (see for instance [72, 88]), much slower than predicted for independent returns. 1, the excess kurtosis (which is zero for a normal distribution) typically remains large even for monthly returns, testifying (i) of signiﬁcant deviations from normality, (ii) of the heavy tail behavior of the distribution of returns and (iii) of signiﬁcant time dependences between asset returns . 2 Mechanisms for Power Laws in Finance Another approach rooted in economic theory, which can be invoked to derive the distribution of ﬁnancial returns, consists in applying the “Gibrat principle”  initially introduced to account for the growth of cities and of wealth through a mechanism combining stochastic multiplicative and additive noises [55, 207, 268, 446, 454] leading to a Pareto distribution of sizes [94, 193].
A well-known result of the perturbation theory states that such coupling leads to a “repulsion” between the eigenvalues, which can be observed in Fig. 2 where C has been constructed with three block matrices A1 , A2 , and A3 and non-zero oﬀ-diagonal coupling described in the ﬁgure caption. These values allow to quantitatively replicate the empirical ﬁnding of Laloux et al. in , where the three ﬁrst eigenvalues are approximately λ1 57, λ2 10 and λ3 8. 5 3 Fig. 2. Spectrum of eigenvalues estimated from the sample correlation matrix of N = 406 time series of length T = 1309.
Extreme Financial Risks: From Dependence to Risk Management (Springer Finance) by D. Sornette, Y. Malevergne