A nested factor model for non-linear dependencies in stock returns
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 [Int. J. Theor. Appl. Finance, 2012, 15]. 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.