  We devise an explicit method to integrate $\alpha$-stable stochastic
differential equations (SDEs) with non-Lipschitz coefficients. To mitigate
against numerical instabilities caused by unbounded increments of the L\'evy
noise, we use a deterministic map which has the desired SDE as its homogenised
limit. Moreover, our method naturally overcomes difficulties in expressing the
Marcus integral explicitly. We present an example of an SDE with a natural
boundary showing that our method respects the boundary whereas Euler-Maruyama
discretisation fails to do so. As a by-product we devise an entirely
deterministic method to construct $\alpha$-stable laws.
