On the multidimensional extension of countermonotonicity and its applications

In a 2-dimensional space, Fréchet-Hoeffding upper and lower bounds define comonotonicity and countermonotonicity, respectively. Similarly, in the multidimensional case, comonotonicity can be defined using the Fréchet-Hoeffding upper bound. However, since the multidimensional Fréchet-Hoeffding lower bound is not a distribution function, there is no obvious extension of countermonotonicity in multidimensions. This paper investigates in depth a new multidimensional extension of countermonotonicity. We first provide an equivalent condition for countermonotonicity in 2-dimension, and extend the definition of countermonotonicity into multidimensions. In order to justify such extensions, we show that newly defined countermonotonic copulas constitute a minimal class of copulas. Two applications will be provided. First, we will study the relationships between multidimensional countermonotonicity and such well-known multivariate concordance measures as Kendall's tau or Spearman's rho. Second, we will give a financial interpretation of multidimensional countermonotonicity via the existing herd behavior index.