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.