  This paper presents simple formulae for the local variance gamma model of
Carr and Nadtochiy, extended with a piecewise-linear local variance function.
The new formulae allow to calibrate the model efficiently to market option
quotes. On a small set of quotes, exact calibration is achieved under one
millisecond. This effectively results in an arbitrage-free interpolation of
class $C^2$. The paper proposes a good regularization when the quotes are
noisy. Finally, it puts in evidence an issue of the model at-the-money, which
is also present in the related one-step finite difference technique of
Andreasen and Huge, and gives two solutions for it.
