  This paper studies randomized quasi-Monte Carlo (QMC) sampling for
discontinuous integrands having singularities along the boundary of the unit
cube $[0,1]^d$. Both discontinuities and singularities are extremely common in
the pricing and hedging of financial derivatives and have a tremendous impact
on the accuracy of QMC. It was previously known that the root mean square error
of randomized QMC is only $o(n^{-1/2})$ for discontinuous functions with
singularities. We find that under some mild conditions, randomized QMC yields
an expected error of $O(n^{-1/2-1/(4d-2)+\epsilon})$ for arbitrarily small
$\epsilon>0$. Moreover, one can get a better rate if the boundary of
discontinuities is parallel to some coordinate axes. As a by-product, we find
that the expected error rate attains $O(n^{-1+\epsilon})$ if the
discontinuities are QMC-friendly, in the sense that all the discontinuity
boundaries are parallel to coordinate axes. The results can be used to assess
the QMC accuracy for some typical problems from financial engineering.
