  In this article we introduce a finite difference approximation for
integro-differential operators of L\'evy type. We approximate solutions of
integro-differential equations, where the second order operator is allowed to
degenerate. In the existing literature, the L\'evy operator is treated as a
zero/first order operator outside of a centered ball of radius $\delta$,
leading to error estimates of order $\xi (\delta)+N(\delta)(h+\sqrt{\tau})$,
where $h$ and $\tau$ are the spatial and temporal discretization parameters
respectively. In these estimates $\xi (\delta) \downarrow 0$, but $N(\delta
)\uparrow \infty$ as $\delta \downarrow 0$. In contrast, we treat the
integro-differential operator as a second order operator on the whole unit
ball. By this method we obtain error estimates of order $(h+\tau^k)$ for $k\in
\{1/2,1\}$, eliminating the additional errors and the blowing up constants.
Moreover, we do not pose any conditions on the L\'evy measure.
