Why is the volatility of single stocks so much rougher than that of the S&P500?

ArXiv ID: 2505.02678 “View on arXiv”

Authors: Othmane Zarhali, Cecilia Aubrun, Emmanuel Bacry, Jean-Philippe Bouchaud, Jean-François Muzy

Abstract

The Nested factor model was introduced by Chicheportiche et al. 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 et al. , where it has been observed that the Hurst exponents of stock 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.

Keywords: nested factor model, rough volatility, Hurst exponent, multifractal processes, volatility modeling, Equity

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper presents advanced stochastic calculus and fractional Brownian motion theory with extensive mathematical derivations and theoretical proofs, indicating high math complexity. It validates its model with numerical experiments, statistical estimation procedures with theoretical guarantees, and empirical analysis on S&P500 data, demonstrating substantial empirical rigor.

  flowchart TD
    Start["Research Goal: Why are single stock volatilities<br>rougher than S&P500 index volatilities?"]

    Method["Methodology: Nested Factor Model<br>with Rough Volatility"] --> Data["Data: S&P500 Daily Stock Returns"]

    Data --> CP["Computational Process:<br>1. Estimate Hurst exponent H<br>2. Analyze Factor vs Residual roughness"]

    CP --> Finding1["Key Finding 1: Factor H ≈ 0.11<br>(Rough, dominant mode)"]
    CP --> Finding2["Key Finding 2: Residual H ≈ 0<br>(Super-rough/Multifractal)"]

    Finding1 & Finding2 --> Conclusion["Outcome: Model explains why<br>index vol > single stock vol"]