On the relationship between modular functions and copulas

Copulas are nothing else but supermodular functions with absorbing element 0 and neutral element 1. Although a copula C cannot be modular on the unit square itself, it is effectively so on every rectangle with zero C-volume. In this paper, we show that an appropriate modular function on the unit square can be transformed into a copula by performing a simple truncation operation (either from below or above to ensure the compatibility with the Fréchet–Hoeffding bounds). Modular functions on the unit square admit an additive representation in terms of two univariate functions. We specifically focus on cases where these univariate functions are sections (horizontal, vertical, diagonal) of a given copula. Ample illustrations are provided.