  In some options markets (e.g. commodities), options are listed with only a
single maturity for each underlying. In others, (e.g. equities, currencies),
options are listed with multiple maturities. In this paper, we provide an
algorithm for calibrating a pure jump Markov martingale model to match the
market prices of European options of multiple strikes and maturities. This
algorithm only requires solutions of several one-dimensional root-search
problems, as well as application of elementary functions. We show how to
construct a time-homogeneous process which meets a single smile, and a
piecewise time-homogeneous process which can meet multiple smiles.
