  Modeling the evolution of a financial index as a stochastic process is a
problem awaiting a full, satisfactory solution since it was first formulated by
Bachelier in 1900. Here it is shown that the scaling with time of the return
probability density function sampled from the historical series suggests a
successful model. The resulting stochastic process is a heteroskedastic,
non-Markovian martingale, which can be used to simulate index evolution on the
basis of an auto-regressive strategy. Results are fully consistent with
volatility clustering and with the multi-scaling properties of the return
distribution. The idea of basing the process construction on scaling, and the
construction itself, are closely inspired by the probabilistic renormalization
group approach of statistical mechanics and by a recent formulation of the
central limit theorem for sums of strongly correlated random variables.
