Multiscaling in the Rough Bergomi Model: A Tale of Tails

Paper: arXiv 2601.11305

Authors: Giuseppe Brandi, Tiziana Di Matteo

Abstract

The rough Bergomi (rBergomi) model, characterised by its roughness parameter $H$, has been shown to exhibit multiscaling behaviour as $H$ approaches zero. Multiscaling has profound implications for financial modelling: it affects extreme risk estimation, influences optimal portfolio allocation across different time horizons, and challenges traditional option pricing approaches that assume uniscaling behaviours. Understanding whether multiscaling arises primarily from the roughness of volatility paths or from the resulting fat-tailed returns has important implications for financial modelling, option pricing, and risk management. This paper investigates the real source of this multiscaling behaviour by introducing a novel two-stage statistical testing procedure. In the first stage, we establish the presence of multiscaling in the rBergomi model against an uniscaling fractional Brownian motion process. We quantify multiscaling by using weighted least squares regression that accounts for heteroscedastic estimation errors across moments. In the second stage, we apply shuffled surrogates that preserve return distributions while destroying temporal correlations. This is done by using distance-based permutation tests robust to asymmetric null distributions. In order to validate our procedure, we check the robustness of the results by using synthetic processes with known multifractal properties, namely the Multifractal Random Walk (MRW) and the Fractional Lévy Stable Motion (FLSM). We provide compelling evidence that multiscaling in the rBergomi model arises primarily from fat-tailed return distributions rather than memory effects. Our findings suggest that the apparent multiscaling in rough volatility models is largely attributable to distributional properties rather than genuine temporal scaling behaviour.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper tackles a key theoretical question in rough volatility models by combining rigorous fractional stochastic calculus with sophisticated statistical techniques like distance-based permutation tests and surrogate data. It presents an innovative, well-explained framework that convincingly isolates fat-tailed returns from memory effects in multiscaling behavior.

Research Flowchart

  flowchart TD
    A["Research Goal: Identify Source of Multiscaling in rBergomi Model (H → 0)"] --> B["Data & Models: rBergomi, FBM, MRW, and FLSM Synthetic Processes"]
    B --> C["Stage 1: Test Multiscaling vs Uniscaling (Weighted Least Squares Regression)"]
    C --> D["Stage 2: Shuffled Surrogates & Distance-Based Permutation Tests"]
    D --> E["Validation: Test Procedure on Synthetic Processes (MRW & FLSM)"]
    E --> F["Key Outcome: Multiscaling Driven by Fat-Tailed Distributions, Not Memory Effects"]

Cited References (65)

1. Volatility is rough (2018) — link
2. Scaling Behaviors in Differently Developed Markets (2003)
3. Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series (2002)
4. Statistical physics and economic fluctuations: do outliers exist? (2003)
5. Measuring Multiscaling in Financial Time-Series (2016)
6. Understanding the Source of Multifractality in Financial Markets (2012)
7. The Components of Empirical Multifractality in Financial Returns (2009)
8. The Origins of Multifractality in Financial Time Series and the Effect of Extreme Events (2014)

1 of 65 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.