Local Variance Gamma - (EXTRA SEMINAR)

In some options markets (eg. commodities), options are listed with only a single maturity for each underlying.

In others, (eg. equities, currencies),

options are listed with multiple maturities.

In this paper, we assume that the risk-neutral process for the underlying futures price is a pure jump Markov martingale and that European option prices are given at a continuum of strikes and at one or more maturities. 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.

We also show that our construction leads to partial differential difference equations (PDDE's), which permit both explicit calibration and fast numerical valuation