  The aim of our work is to propose a natural framework to account for all the
empirically known properties of the multivariate distribution of stock returns.
We define and study a "nested factor model", where the linear factors part is
standard, but where the log-volatility of the linear factors and of the
residuals are themselves endowed with a factor structure and residuals. We
propose a calibration procedure to estimate these log-vol factors and the
residuals. We find that whereas the number of relevant linear factors is
relatively large (10 or more), only two or three log-vol factors emerge in our
analysis of the data. In fact, a minimal model where only one log-vol factor is
considered is already very satisfactory, as it accurately reproduces the
properties of bivariate copulas, in particular the dependence of the
medial-point on the linear correlation coefficient, as reported in
Chicheportiche and Bouchaud (2012). We have tested the ability of the model to
predict Out-of-Sample the risk of non-linear portfolios, and found that it
performs significantly better than other schemes.
