What are copulas?

Answer 1

Copulas are an exremely useful tool used to build models of the joint behavior of multiple financial variables. They allow you to define a multivariate statistical distribution in two steps:

1. first you specify the marginal univariate distribution for each of the variables of interest
2. then you link the single univariate distributions via a copula in order to obtain a multivariate distribution

Basically, what the copula does is to specify the structure of the dependence among the variables, leaving their marginal distributions unaltered. Mathematically, a copula is a scalar-valued function of n-variables. If you plug n univariate distribution functions into its n arguments you get a multivariate distribution function, which has the original n distribution functions as its marginals. Stated more formally (for the case of two variables), if C=C(u,v) is the copula and F(x) and G(y) are two univariate distribution functions, H=H(x,y)=C(F(x),G(y)) is a bivariate distribution function having C and G as its marginals.

Answer 2

Copulas are statistical functions that describe the relationship between multiple variables by capturing the dependency structure between them. They are commonly used in finance, insurance, and risk management to model complex dependencies that cannot be captured by traditional correlation measures. Copulas provide a flexible way to model and simulate multivariate data, even when the variables have different distributions.