A simple model for simulating cross correlations of a many-assets market is discussed. Correlations between assets are initially considered within the context of the well-known one-factor model, in which a driving term common to all stocks is present. The results are compared to those of real market data corresponding to a set of 445 stocks taken from the Standard and Poors 500 index. The model is further extended by introducing a stochastic volatility within each time series using an autoregressive scheme. This artificial market reproduces the empirically observed fat tails in the distribution function of logarithmic price variations and, more important, leads to additional cross correlations between the time series, in better agreement with the real market behavior.