Extreme correlation
In this article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2024) explains the concept of extreme correlation.
Background
In financial risk management, there is a concept that is often overlooked, the extreme correlation also known as tail dependence. Tail dependence reveals how extreme events in two variables are linked. The oversight could leave portfolios exposed to amplified risks during market turbulence. In this post, we will get to see the definition and implications of this concept.
Linear correlation and copula
As presented in the post on copula, using linear correlation to model the dependence structure between random variables poses many limitations, and copula is a more generalized tool that allows to capture a fuller picture of the dependence structure.
Let’s recall the definition of copula. A copula, denoted typically as C∶[0,1]d→[0,1] , is a multivariate distribution function whose marginals are uniformly distributed on the unit interval. The parameter d is the number of variables. For a set of random variables U1, …, Ud with cumulative distribution functions F1, …, Fd, the copula function C satisfies:
C(F1(u1),…,Fd(ud)) = ℙ(U1≤u1,…,Ud≤ud)
Here we introduce Student t-copula as an example, which will also be used as an illustration in the part of extreme correlation.
Tail dependence coefficient
The tail dependence coefficient captures the dependence level of a bivariate distribution at its tails. Let’s denote X and Y as two continuous random variables with continuous distribution F and G respectively. The (upper) tail dependence coefficient between X and Y is defined as:
with the limit of λU∈[0,1]
We can conclude that the tail dependence coefficient between two continuous random variables is a copula property, and it remains invariant with strict increasing transformations of the two random variables.
If λU∈(0,1], X and Y are considered asymptotically dependent in their (upper) tail. If λU=0, X and Y are considered asymptotically independent in their (upper) tail.
It is important to note that the independent of X and Y implies that λU=0, but the converse is not necessarily true. λU describes only the dependence level at the tails.
Examples of extreme correlation
Longin and Solnik (2001) and Gkillas and Longin (2019) employ the logistic model for the dependence function of the Gumbel copula (also called the Gumbel-Hougaard copula) for Fréchet margins, as follows:
This model contains the special cases of asymptotic independence and total dependence. It is parsimonious, as we only need one parameter to model the bivariate dependence structure of exceedances, i.e., the dependence parameter α with 0<α≤1. The correlation of exceedances ρ (also called extreme correlation) can be computed from the dependence parameter α of the logistic model as follows: ρ= 1-α^2. The special cases where α is equal to 1 and α converges towards 0 correspond to asymptotic independence, in which ρ is equal to 0, and total dependence, in which ρ is equal to 1, respectively (Tiago de Oliveira, 1973).
Related posts on the SimTrade blog
About extreme value theory
▶ Shengyu ZHENG Extreme Value Theory: the Block-Maxima approach and the Peak-Over-Threshold approach
▶ Shengyu ZHENG Optimal threshold selection for the peak-over-threshold approach of extreme value theory
▶ Gabriel FILJA Application de la théorie des valeurs extrêmes en finance de marchés
Useful resources
Academic resources
Gkillas K. and F. Longin (2018) Is Bitcoin the new digital Gold?, Working paper, ESSEC Business School.
Longin F. (2016) Extreme events in finance: a handbook of extreme value theory and its applications Wiley Editions.
Longin F. and B. Solnik (2001) Extreme Correlation of International Equity Markets, The Journal of Finance, 56, 649-676.
Zeevi A. and R. Mashal (2002) Beyond Correlation: Extreme Co-Movements between Financial Assets. Available at SSRN: https://ssrn.com/abstract=317122
Other resources
Rieder H. E. (2014) Extreme Value Theory: A primer (slides).
About the author
The article was written in January 2024 by Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2024).