  We consider the integration of two-dimensional, piecewise constant functions
with respect to copulas. By drawing a connection to linear assignment problems,
we can give optimal upper and lower bounds for such integrals and construct the
copulas for which these bounds are attained. Furthermore, we show how our
approach can be extended in order to approximate extremal values in very
general situations. Finally, we apply our approximation technique to problems
in financial mathematics and uniform distribution theory, such as the
model-independent pricing of first-to-default swaps.
