Rough Bergomi Volatility Process

• Rough Bergomi is a stochastic volatility model driven by fractional noise (H < 1/2) that creates rough sample paths and captures short-maturity implied volatility smiles.
• The model provides closed-form expressions for forward variance curves, facilitating the pricing of VIX and SPX derivatives through efficient numerical schemes.
• A joint calibration methodology aligns market data from VIX futures and SPX options, though discrepancies like a 20% vol-of-vol difference highlight real-world challenges.

The rough Bergomi volatility process is a stochastic volatility model where the instantaneous variance is driven by a fractional process with Hurst parameter $H<\frac12$, producing "rough" sample paths. This construction permits the accurate fitting of short-maturity implied volatility smiles and facilitates market-consistent calibration between equity (SPX) and volatility (VIX) derivatives. The process is mathematically tractable, supports efficient numerical algorithms, and encapsulates key stylized facts observed in financial markets, such as steep at-the-money skews and term structure of implied volatility.

1. Mathematical Formulation of the Rough Bergomi Process

The rough Bergomi (rBergomi) process models the log-stock price $X_t = \log S_t$ and its instantaneous variance $V_t$ using dynamics driven by fractional noise. The dynamics are given by

$$dX_t = -\frac12 V_t\,dt + \sqrt{V_t}\,dW_t, \qquad V_t = \xi_0(t)\,\mathcal{E}(2\nu C_H\,\mathcal{V}_t),$$

where:

• $\xi_0(t)$ is the initial forward variance curve, considered a market observable (often extracted from variance swaps).
• $\nu$ is the volatility-of-volatility parameter.
• $C_H$ is a normalization constant:

$$C_H = \sqrt{ \frac{2H\;\Gamma(3/2-H)}{ \Gamma(H+1/2)\;\Gamma(2-2H)} }$$

• $\mathcal{E}(\cdot)$ denotes the Doléans-Dade stochastic exponential.
• The Volterra process $\mathcal{V}_t$ is defined by

$\mathcal{V}_t^{(T)} = \int_0^T (t-s)^{H-1/2} dZ_s,$

where $Z$ is a Brownian motion, possibly correlated with $W$ (the driver of the stock).

The fractional kernel $(t-s)^{H-1/2}$, $H<1/2$, endows the variance process with persistent, low-Hölder regularity, resulting in empirical agreement with observed volatility trajectories and option smiles.

A key closed-form for the future forward variance curve, crucial for VIX pricing and calibration, is: $$\xi_T(t) = \xi_0(t)\,\eta_T(t)\,\exp\left\{ \frac{\nu^2 C_H^2}{H}\big[ (t-T)^{2H} - t^{2H} \big] \right\}, \;\; t\geq T.$$

2. VIX Derivatives and Forward Variance Curve Evolution

In this framework, the VIX squared at time $T$ is given (in the risk-neutral measure) by

$${\rm VIX}_T^2 = \frac{1}{\Delta} \int_{T}^{T+\Delta} \xi_T(s)\,ds,$$

where $\Delta$ denotes the standardized VIX window (e.g., 30 days expressed in years).

The price of a VIX future at time $0$ maturing at $T$ is

$$\mathcal{V}_T = \mathbb{E}[ {\rm VIX}_T \mid \mathcal{F}_0] = \mathbb{E}\left[ \sqrt{ \frac{1}{\Delta} \int_{T}^{T+\Delta} \xi_T(s)\,ds } \, \middle|\, \mathcal{F}_0 \right].$$

This structure exploits formula (6) above to express all future variance (and thus VIX) dynamics as a transformation of $\xi_0$ and the Volterra process increments, allowing the entire forward variance curve to evolve in a tractable, explicit manner. Conditional Gaussianity is used to obtain further approximations and efficient estimators for VIX futures and options, including lognormal-type closed-form expressions.

3. Simulation and Pricing Algorithms

Simulation of the rough Bergomi process is challenging due to the non-Markovian, fractional kernel. Two main approaches are described:

• Hybrid scheme (Bennedsen, Lunde, Pakkanen): The Volterra fractional process is simulated approximately via the discretized convolution, with complexity $O(n\log n)$, suitably capturing kernel singular behavior near zero.
• Truncated Cholesky decomposition: For high-precision, short-horizon tasks (e.g., VIX computations on a 30-day window), truncated Cholesky sampling of the covariance matrix is possible, but numerical instability may arise for very fine discretizations.

After simulating the underlying Volterra process, the variance trajectory is propagated forward, and pricing of options (e.g., VIX or SPX) is performed using forward Euler discretization, with integrals over [T, T+Δ] for VIX evaluated via quadrature.

Lognormal approximations can be applied for integrated variance over the VIX window: $$V_T \approx \Delta^{-1/2}\sqrt{\int_T^{T+\Delta}\xi_0(t)\,dt}\ \exp(-\sigma^2/8)$$ with $\sigma^2$ computed from the first two moments of the log-averaged variance.

4. Joint Calibration Methodology

Calibration is executed in two steps:

1. VIX Futures calibration: Optimize parameters $(\nu, H)$ and the initial forward variance curve $\xi_0(t)$ to fit observed VIX futures prices, using the efficient pricing formulae above.
2. SPX options calibration: With $(H, \xi_0)$ held fixed (as roughness is considered universal), adjust $(\nu, \rho)$—including the Brownian correlation—by least-squares fit to SPX option prices, employing Monte Carlo simulation per Algorithm 4.1.

This strategy leverages the eSSVI parameterization for extracting $\xi_0(t)$ from SPX options, thus utilizing the full term structure of ATM variance (and its time derivative) as market input. Ultimately this produces consistent calibration across both volatility (VIX) and equity (SPX) derivatives.

Calibration Step	Parameters Fixed	Parameters Fitted	Market Data Used
1. VIX futures	—	$(\nu, H, \xi_0(t))$	VIX futures prices
2. SPX options	$(H, \xi_0)$	$(\nu, \rho)$	SPX option prices

5. Theoretical Implications: Volatility Smile and Roughness

Utilization of rough volatility (fractional kernel with $H<1/2$) allows the model to explain observed phenomena such as:

• Steep short-maturity implied volatility smile: The non-Markovian, persistent structure of the volatility driver causes pronounced at-the-money skews for short-dated options, in line with empirical data.
• Persistence in variance: The low-Hölder regularity and path dependence of the variance process generate both short-range volatility clustering and long-range correlation decay properties.

The lognormal approximation for the VIX, enabled by the explicit forward variance formula (6), delivers high numerical accuracy in pricing (errors $\sim 10^{-3}$) and computational efficiency superior to fully simulated schemes.

Notably, joint calibration exposes possible market inconsistencies, such as a 20% discrepancy in fitted vol-of-vol between VIX and SPX products. This signals real-world deviations between implied variance dynamics and observed forward variance surfaces, meriting deeper investigation.

6. Applicability and Practical Performance

The rough Bergomi process offers several practical benefits:

• Model flexibility: The explicit link between the forward variance curve at time $T$ and the market-observable initial curve $\xi_0$ allows for easy incorporation of market information.
• Efficient pricing: The semi-closed and lognormal approximations for VIX derivatives avoid full pathwise simulation.
• Calibration consistency: The two-stage algorithm ensures model parameters provide a unified explanation for both VIX and SPX markets.
• Numerical tractability: The hybrid and Cholesky simulation methods balance speed and accuracy, enabling use in real trading and risk management contexts.

The model outperforms classical Markovian stochastic volatility approaches in matching the steepness of short-maturity option skews, although it may still display calibration discrepancies when aligning disparate markets.

7. Limitations and Further Directions

While the rough Bergomi process accurately captures many features of volatility and option prices, notable limitations persist:

• The non-Markovian nature entails greater computational complexity, with some schemes suffering from ill-conditioning for very fine discretizations.
• Calibration may indicate structural model disagreement between SPX and VIX, reflected in non-uniform parameter fits.
• The framework is best suited for settings where the entire forward variance curve is reliably observable or can be robustly extracted.
• Full understanding of market discrepancies and extension to more general rough or multifactor models remain important avenues for future work.

In summary, the rough Bergomi volatility process provides a well-posed, empirically validated stochastic volatility framework that leverages fractional noise to explain the term structure and steepness of implied volatility smiles at short maturities. Its mathematical tractability and explicit formulae for forward variance curves facilitate scalable pricing, calibration, and risk assessment across major equity and volatility derivatives markets (Jacquier et al., 2017).