  We investigate methods for pricing American options under the variance gamma
model. The variance gamma process is a pure jump process which is constructed
by replacing the calendar time by the gamma time in a Brownian motion with
drift, which makes it a time-changed Brownian motion. In general, the finite
difference method and the simulation method can be used for pricing under this
model, but their speed is not satisfactory. So there is a need for fast but
accurate approximation methods. In the case of Black-Merton-Scholes model,
there are fast approximation methods, but they cannot be utilized for the
variance gamma model. We develop a new fast method inspired by the quadratic
approximation method, while reducing the error by making use of a machine
learning technique on pre-calculated quantities. We compare the performance of
our proposed method with those of the existing methods and show that this
method is efficient and accurate for practical use.
