  We introduce an algorithm for the pricing of finite expiry American options
driven by L\'evy processes. The idea is to tweak Carr's `Canadisation' method,
cf. Carr [9] (see also Bouchard et al [5]), in such a way that the adjusted
algorithm is viable for any L\'evy process whose law at an independent,
exponentially distributed time consists of a (possibly infinite) mixture of
exponentials. This includes Brownian motion plus (hyper)exponential jumps, but
also the recently introduced rich class of so-called meromorphic L\'evy
processes, cf. Kyprianou et al [16]. This class contains all L\'evy processes
whose L\'evy measure is an infinite mixture of exponentials which can generate
both finite and infinite jump activity. L\'evy processes well known in
mathematical finance can in a straightforward way be obtained as a limit of
meromorphic L\'evy processes. We work out the algorithm in detail for the
classic example of the American put, and we illustrate the results with some
numerics.
