Which means that random variables \(U_{1},...,U_{d}\) are independent if and only if their Copula is \(\Pi\). Comonotonicity copula is the upper bound of every copula function mentioned in equation (2). It represents perfectly positive dependence structure between random variables. While perfectly negative dependence structure is described by the lower Frechet bound in (2) and called countermonotonicity copula.

Implicit Copulas

One of the most popular representatives of implicit copulas is Gaussian copula, which was introduced in valuation of financial derivatives as Credit Swaps, First-to-Default Contracts by Li (1999). Being widely adopted by finance professionals due to its simplicity before the Financial Crisis in 2008, later it was accused as "The formula that killed Wall Street" by Felix Salmon. General representation of Gaussian copula is as follows: