  In the present paper we construct stock price processes with the same
marginal log-normal law as that of a geometric Brownian motion and also with
the same transition density (and returns' distributions) between any two
instants in a given discrete-time grid. We then illustrate how option prices
based on such processes differ from Black and Scholes', in that option prices
can be either arbitrarily close to the option intrinsic value or arbitrarily
close to the underlying stock price. We also explain that this is due to the
particular way one models the stock-price process in between the grid time
instants which are relevant for trading. The theoretical result concerning
scalar stochastic differential equations with prescribed diffusion coefficient
whose densities evolve in a prescribed exponential family, on which part of the
paper is based, is presented in detail.
