Copulas are constructs of probability theory that have attracted increased attention in the past few decades. In only the past few years, financial professionals have turned to them for modeling high-dimensional problems, such as value-at-risk or portfolio credit risk.

Statistical and time series methods used in finance focus on estimating marginal distributions for the components of random vectors—as opposed to estimating entire joint distributions. A problem with this is the fact that marginal distributions do not define a joint distribution. Indeed, given a set of marginal distributions, there are infinitely many joint distributions that can be used to couple them. Suppose a Monte Carlo analysis needs to be performed with a random vector that has normal, lognormal, Student-t and uniform marginals. How should realizations be generated? The answer depends upon what joint distribution is assumed. Copulas provide a means of specifying such an assumption.

For an n-dimensional vector for which only marginal distributions have been specified, a copula is a function from the n-dimensional cube to the real numbers. It specifies a joint distribution for the random vector that is consistent with all the marginals. In finance, copulas are useful for implementing Monte Carlo analyses, as indicated above, and for specifying non-linear dependencies between random variables—which linear correlations do not capture.