A diffusive model with a price dependent diffusion coefficient was recently proposed to explain the occurrence of non-Gaussian price return distributions observed empirically in real markets
[J.L. McCauley and G.H. Gunaratne, Physica A 329, 178 (2003)].
Depending on the functional form of the diffusion coefficient, the exactly solved continuum limit of the model can produce either an exponential distribution, or a "fat-tailed" power-law distribution of returns. Real markets, however, are discrete, and, in this paper, the effects of discreteness on the model are explored. Discrete distributions from simulations and from numerically exact calculations are presented and compared to the corresponding distributions of the continuum model. A type of phase transition is discovered in discrete models that lead to fat-tailed distributions in the continuum limit, sheading light on the nature of such distributions. The transition is to a phase in which infinite price changes can occur in finite time.