A new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices. The unit period distribution is normal conditional on a variance that is distributed as a gamma variate. Its advantages include long tailedness, continuous-time specification, finite moments of all orders, elliptical multivariate unit period distributions, and good empirical fit. The process is pure jump, approximable by a compound Poisson process with high jump frequency and low jump magnitudes. Applications to option pricing show differential effects for options on the money, compared to in or out of the money. Copyright 1990 by the University of Chicago.