Multiscaling in the Rough Bergomi Model: A Tale of Tails

TL;DR

This paper investigates the source of multiscaling in the rough Bergomi model by contrasting fat-tailed return distributions with temporal memory through a two-stage surrogate data framework. It introduces a Generalised Hurst Exponent approach to quantify multiscaling via $H(q)$ and a multiscaling proxy $B$, and uses matched fractional Brownian motion and shuffled surrogates to separate distributional and memory effects. The main finding is that multiscaling in the rBergomi model is largely due to fat-tailed returns, especially for very rough regimes with $H\to 0$, while temporal dependencies become more prominent only at moderate roughness ($H\gtrsim 0.1$). The methodology is validated on synthetic models (MRW and FLSM) and shown to robustly attribute sources of multiscaling, with practical implications for risk management and option pricing in rough volatility contexts.

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.

Figures (4)

• Figure 1: Rough Bergomi model characteristics for different values of $H$. Top row: Simulated price paths $S_t$ over 10000 observations. Middle row: Discrete returns $r_t = S_t - S_{t-1}$ computed from the price process. Bottom row: Realised volatility $\sqrt{v_t}$ where $v_t$ is the instantaneous variance process. Parameters: $\xi_0 = 0.1$ (initial variance), $\rho = -0.9$ (correlation between returns and volatility), $\eta = 1.9$ (volatility of volatility). As $H$ decreases, price paths become more irregular, return distributions develop heavier tails, and volatility exhibits more pronounced clustering and roughness.
• Figure 2: Multiscaling proxy $B$ as a function of $H$ in the rough Bergomi model.
• Figure 3: Kurtosis of returns (log scale) in the rough Bergomi model as a function of the Hurst parameter $H$. Tail heaviness decreases substantially as $H$ increases, with median values dropping from around 50 for very rough processes ($H = 0.001$) to approximately 5 for $H = 0.2$. Each boxplot summarises 1000 independent simulations.
• Figure 4: Volatility clustering in the rough Bergomi model as a function of the Hurst parameter $H$, measured as the sum of the first 10 autocorrelation coefficients of absolute returns. Temporal persistence increases dramatically with $H$, rising from near zero for very rough processes ($H \leq 0.01$) to approximately 3 for $H = 0.2$. Each boxplot summarises 1000 independent simulations.