2.2.2 Copulas<br/>&nbsp; When the two variables are independent, the joint density is simply the product<br/>of the marginal densities. It is rarely the case, however, that financial variables are<br/>independent. Dependencies can be modeled by a function called the copula, which<br/>links, or attaches, marginal distributions into a joint distribution. Formally, the<br/>copula is a function of the marginal distributions F (x), plus some parameters, θ,<br/>that are specific to this function (and not to the marginals). In the bivariate case,<br/>it has two arguments:<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; c12[F1(x1), F2(x2); θ] (2.20)<br/>The link between the joint and marginal distribution is made explicit by Sklar’s<br/>theorem, which states that, for any joint density, there exists a copula that links<br/>the marginal densities<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; f12(x1, x2) = f1(x1) × f2(x2) × c12[F1(x1), F2(x2); θ] (2.21)<br/>With independence, the copula function is a constant always equal to one.<br/>Thus the copula contains all the information on the nature of the dependence<br/>between the random variables but gives no information on the marginal distributions.<br/>Complex dependencies can be modeled with different copulas. Copulas are<br/>now used extensively for modeling financial instruments such as collateralized debt<br/>obligations (CDOs). As we shall see in a later chapter, CDOs involve movements<br/>in many random variables, which are the default events for the companies issuing<br/>the debt.