A nested factor model for non-linear dependences in stock returns

TL;DR

This paper introduces a nested factor model for stock returns that captures non-linear dependencies by modeling log-volatility factors, improving risk prediction for non-linear portfolios.

Contribution

It proposes a novel nested factor model with a calibration method, revealing that few log-volatility factors suffice to replicate complex dependence structures in stock returns.

Findings

Few log-volatility factors (2-3) are sufficient to model dependencies.

The minimal model accurately reproduces bivariate copula properties.

The model outperforms existing schemes in out-of-sample risk prediction.

Abstract

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.