Rough Volatility Fractional Brownian Motion Gatheral, Jaisson & Rosenbaum (2018)

Rough Volatility & Bergomi Lab

Simulate Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum's (2018) "Volatility is Rough" framework. Model Riemann-Liouville fractional Brownian motion, sub-diffusive Hurst parameters ($H < 0.5$), Rough Bergomi (rBergomi) variance dynamics, and short-dated ATM implied volatility skew explosion.

Theoretical Foundations: Fractional Brownian Motion, Rough Bergomi & ATM Skew Scaling
1. Riemann-Liouville Fractional Brownian Motion ($W_t^H$)

Unlike standard Brownian motion with independent increments, fractional Brownian motion with Hurst parameter $H \in (0, 1)$ possesses memory. The Riemann-Liouville representation is given by:

$$W_t^H = \sqrt{2H} \int_0^t (t - s)^{H - 1/2} dW_s, \quad b_j = \frac{(j+1)^{H+1/2} - j^{H+1/2}}{H + 1/2}$$

When $H < 1/2$, the kernel $(t-s)^{H-1/2}$ is singular at $s \to t$, creating paths that are far more irregular and jagged than standard Brownian diffusion ($H = 1/2$).

2. Rough Bergomi Variance & ATM Skew Explosion

The spot variance $v_t$ is modeled as a log-normal fractional martingale driven by vol-of-vol $\eta$ and correlated with asset returns via $d\langle W^S, W \rangle_t = \rho dt$:

$$v_t = \xi_0 \exp\left(\eta W_t^H - \frac{1}{2}\eta^2 t^{2H}\right), \quad \psi(\tau) = \left|\frac{\partial \sigma_{\text{IV}}}{\partial \ln K}\right|_{\text{ATM}} \approx \frac{|\rho|\eta}{2(H + 1/2)}\tau^{H - 1/2}$$

As maturity $\tau \to 0$, classical diffusion ($H = 0.5$) yields a flat skew $\tau^0 = \text{const}$. In contrast, rough volatility ($H \approx 0.12$) produces a steep power-law explosion $\tau^{-0.38}$, perfectly matching real options data.

Model Parameters

SPX Rough Equity Index: Calibrated to US equity index volatility surfaces (H = 0.12, ρ = -0.85). Exhibits extreme short-dated implied volatility skew explosion (τ^(H - 0.5) ≈ τ^-0.38) matching observed options data.

Hurst Parameter $H$ 0.12 Ultra-Rough (H < 0.2)
7-Day ATM Skew $\psi(7\text{d})$ -- Power-law: $\tau^{H - 1/2}$
Skew Exponent $H - \frac{1}{2}$ -0.38 Classical: $0.00$
Empirical Hurst $H_{\text{emp}}$ -- Structure Function
Final Price: --
Final Vol: --
Peak Vol: --
Vol-of-Vol $\eta$: --
Dual 5×5 Rough Volatility Sensitivity Grids
Sensitivity Grid 1: Hurst $H$ vs Vol-of-Vol $\eta$ → 7-Day ATM Skew $\psi(7\text{d})$

Demonstrates the power-law explosion in short-dated implied volatility skew as roughness increases ($H \to 0$):

Sensitivity Grid 2: Correlation $\rho$ vs Hurst $H$ → Risk-Neutral Asymmetry Skew

Quantifies how equity leverage correlation and sub-diffusive paths compound implied distribution skewness:

Quantitative Volatility Mastery: Gatheral et al. (2018) Rough Volatility
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Frequently Asked Questions: Rough Volatility & The Rough Bergomi Model

Rough Volatility is a revolutionary paradigm established by Gatheral, Jaisson, and Rosenbaum (2018) demonstrating that log-volatility behaves not as standard Brownian motion (Hurst parameter $H = 0.5$), but as a sub-diffusive fractional Brownian motion with $H \approx 0.10 - 0.15$. Unlike classical models (Black-Scholes, Heston, SABR) where volatility paths are Hölder continuous with exponent $1/2$, rough volatility trajectories are jagged and Hölder continuous with exponent $H < 1/2$.

The Hurst parameter $H \in (0, 1)$ governs the smoothness and memory of a stochastic process. For $H = 0.5$, the process is standard Brownian motion with uncorrelated increments. When $H < 0.5$ (rough regime), increments are negatively correlated, generating highly irregular, anti-persistent, rough trajectories. When $H > 0.5$, increments are positively correlated and persistent (long memory).

In equity options markets, the at-the-money (ATM) implied volatility skew $|\partial \sigma_{\text{IV}} / \partial \ln K|$ explodes as time-to-expiry $\tau$ approaches zero according to a power law $\tau^{H - 1/2}$. Classical models with $H = 0.5$ predict $\tau^0 = \text{const}$ (a flat or bounded skew at short maturities). Setting $H \approx 0.12$ perfectly captures the steep $\tau^{-0.38}$ power-law explosion observed across empirical option surfaces.

The Rough Bergomi model is a non-Markovian variance curve model where spot variance $v_t$ is formulated as $v_t = \xi_0(t) \exp(\eta W_t^H - \frac{1}{2}\eta^2 t^{2H})$, with $W_t^H$ being a Riemann-Liouville fractional Brownian motion, $\eta$ the volatility of volatility, and $\xi_0(t)$ the forward variance curve. It calibrates both the term structure of ATM skew and the volatility smile with only 3 core parameters.

Riemann-Liouville fractional Brownian motion is represented as $W_t^H = \sqrt{2H} \int_0^t (t - s)^{H - 1/2} dW_s$. In discrete time, this is simulated using exact kernel convolution: $W_{t_n}^H = \sqrt{2H} (\Delta t)^H \sum_{j=0}^{n-1} b_j Z_{n - 1 - j}$, where $b_j = \frac{(j + 1)^{H + 1/2} - j^{H + 1/2}}{H + 1/2}$ and $Z$ are standard Gaussian random variables.