Computing the Volume of N-Dimensional Copulas

A problem that is very relevant in applications of copula functions to finance is computation of the survival copula, which in finance is applied to enforce multi-variate put-call parity. This may be very involved for large dimensions. Actually, the problem is a special case of the more general problem of volume computation in high dimensional copulas. Here we provide an algorithm for exact computation of the volume of copula functions, in cases in which the copula function is computable in closed form. We apply the algorithm to the problem of computing the survival copula of a copula function in the pricing problem of a multivariate digital option, and we provide evidence that this is feasible for baskets of up to 20 underlying assets, with acceptable CPU time performances.