Local Variance Gamma and Explicit Calibration to Option Prices

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.