A nested factor model for equity markets: reconciling multifractal stock returns and rough index volatilities

The Nested factor model was introduced by Chicheportiche\textit{ et al.} in \cite{chicheportiche2015nested} to represent non-linear correlations between stocks. Stock returns are explained by a standard factor model, but the (log)-volatilities of factors and residuals are themselves decomposed into factor modes, with a common dominant volatility mode affecting both market and sector factors but also residuals. Here, we consider the case of a single factor where the only dominant log-volatility mode is rough, with a Hurst exponent $H \simeq 0.11$ and the log-volatility residuals are ``super-rough'' or ``multifractal'', with $H \simeq 0$. We demonstrate that such a construction naturally accounts for the somewhat surprising stylized fact reported by Wu \textit{ et al.} in \cite{wu2022rough}, where it has been observed that the Hurst exponents of stock {\it indexes} are large compared to those of individual stocks. We propose a statistical procedure to estimate the Hurst factor exponent from the stock returns dynamics together with theoretical guarantees of its consistency. We demonstrate the effectiveness of our approach through numerical experiments and apply it to daily stock data from the S\&P500 index. The estimated roughness exponents for both the factor and idiosyncratic components validate the assumptions underlying our model.