  We consider L\'evy processes that are approximated by compound Poisson
processes and, correspondingly, BSDEs driven by L\'evy processes that are
approximated by BSDEs driven by their compound Poisson approximations. We are
interested in the rate of convergence of the approximate BSDEs to the ones
driven by the L\'evy processes. The rate of convergence of the L\'evy processes
depends on the Blumenthal--Getoor index of the process. We derive the rate of
convergence for the BSDEs in the $\mathbb L^2$-norm and in the Wasserstein
distance, and show that, in both cases, this equals the rate of convergence of
the corresponding L\'evy process, and thus is optimal.
