  We consider periodic homogenization problems for the L{\'e}vy operators with
asymmetric L{\'e}vy densities. The formal asymptotic expansion used for the
$\a$-stable (symmetric) L{\'e}vy operators ($\a\in (0,2)$) is not applicable
directly to such asymmetric cases. We rescale the asymmetric densities, extract
the most singular part of the measures, which average out the microscopic
dependences in the homogenization procedures. We give two conditions (A) and
(B), which characterize such a class of asymmetric densities, that the above
"rescaled" homogenization is available.
