English

Investigating Extreme Dependences: Concepts and Tools

Statistical Mechanics 2008-12-10 v1 Statistical Finance

Abstract

We investigate the relative information content of six measures of dependence between two random variables XX and YY for large or extreme events for several models of interest for financial time series. The six measures of dependence are respectively the linear correlation ρv+\rho^+_v and Spearman's rho ρs(v)\rho_s(v) conditioned on signed exceedance of one variable above the threshold vv, or on both variables (ρu\rho_u), the linear correlation ρvs\rho^s_v conditioned on absolute value exceedance (or large volatility) of one variable, the so-called asymptotic tail-dependence λ\lambda and a probability-weighted tail dependence coefficient λˉ{\bar \lambda}. The models are the bivariate Gaussian distribution, the bivariate Student's distribution, and the factor model for various distributions of the factor. We offer explicit analytical formulas as well as numerical estimations for these six measures of dependence in the limit where vv and uu go to infinity. This provides a quantitative proof that conditioning on exceedance leads to conditional correlation coefficients that may be very different from the unconditional correlation and gives a straightforward mechanism for fluctuations or changes of correlations, based on fluctuations of volatility or changes of trends. Moreover, these various measures of dependence exhibit different and sometimes opposite behaviors, suggesting that, somewhat similarly to risks whose adequate characterization requires an extension beyond the restricted one-dimensional measure in terms of the variance (volatility) to include all higher order cumulants or more generally the knowledge of the full distribution, tail-dependence has also a multidimensional character.

Keywords

Cite

@article{arxiv.cond-mat/0203166,
  title  = {Investigating Extreme Dependences: Concepts and Tools},
  author = {Y. Malevergne and D. Sornette},
  journal= {arXiv preprint arXiv:cond-mat/0203166},
  year   = {2008}
}

Comments

46 pages including 9 figures

R2 v1 2026-07-22T10:34:53.804Z