  In this paper, we are interested in continuous time models in which the index
level induces some feedback on the dynamics of its composing stocks. More
precisely, we propose a model in which the log-returns of each stock may be
decomposed into a systemic part proportional to the log-returns of the index
plus an idiosyncratic part. We show that, when the number of stocks in the
index is large, this model may be approximated by a local volatility model for
the index and a stochastic volatility model for each stock with volatility
driven by the index. This result is useful in a calibration perspective : it
suggests that one should first calibrate the local volatility of the index and
then calibrate the dynamics of each stock. We explain how to do so in the
limiting simplified model and in the original model.
