  The paper proposes an expanded version of the Local Variance Gamma model of
Carr and Nadtochiy by adding drift to the governing underlying process. Still
in this new model it is possible to derive an ordinary differential equation
for the option price which plays a role of Dupire's equation for the standard
local volatility model. It is shown how calibration of multiple smiles (the
whole local volatility surface) can be done in such a case. Further, assuming
the local variance to be a piecewise linear function of strike and piecewise
constant function of time this ODE is solved in closed form in terms of
Confluent hypergeometric functions. Calibration of the model to market smiles
does not require solving any optimization problem and, in contrast, can be done
term-by-term by solving a system of non-linear algebraic equations for each
maturity, which is fast.
