Sampling from Archimedean copulas

We develop sampling algorithms for multivariate Archimedean copulas. For exchangeable copulas, where there is only one generating function, we first analyse the distribution of the copula itself, deriving a number of integral representations and a generating function representation. One of the integral representations is related, by a form of convolution, to the distribution whose Laplace transform yields the copula generating function. In the infinite-dimensional limit there is a direct connection between the distribution of the copula value and the inverse Laplace transform. Armed with these results, we present three sampling algorithms, all of which entail drawing from a one-dimensional distribution and then scaling the result to create random deviates distributed according to the copula. We implement and compare the various methods. For more general cases, in which an N-dimensional Archimedean copula is given by N-1 nested generating functions, we present algorithms in which each new variate is drawn conditional only on the value of the copula of the previously drawn variates. We also discuss the use of composite nested and exchangeable copulas for modelling random variates with a natural hierarchical structure, such as ratings and sectors for obligors in credit baskets.