Deriving Forward Variance Dynamics in the Rough Bergomi Model

Summary

The document asks for intuition behind a forward variance stochastic differential equation used in rough volatility research. The displayed dynamics depend on the forward variance level, a volatility parameter, the Hurst parameter, the time to the forward maturity, and a Brownian motion. The question itself does not derive the expression or provide a worked calculation.

The answer sketches the modeling route: begin with Bergomi forward variance dynamics, restrict the driver to a single factor, and replace the exponential kernel with a Riemann–Liouville fractional Brownian motion kernel. This connects the forward variance specification to rough volatility through the choice of kernel. The response is brief and directs readers to lecture slides for further intuition; it does not explain the derivation’s assumptions in detail, compare alternative kernels, or discuss calibration or empirical evidence. The excerpt therefore offers a conceptual starting point rather than a complete derivation.

Key ideas

• The question concerns the stochastic dynamics of forward variance in rough Bergomi models.
• The answer starts from Bergomi forward variance dynamics and assumes a single driving factor.
• Replacing an exponential kernel with a Riemann–Liouville fractional Brownian motion kernel yields the rough-process connection.
• The brief explanation does not provide a full derivation or discuss calibration.

Tags

# forward variances under rough bergomi


# forward variances under rough bergomi












I have seen in several papers on rough volatility using the following expression for the forward variances

$$ d\xi_t(u) = \xi_t(u) \eta \sqrt{2H} (u-t)^{H-1/2}dW_t $$

Can anyone explain to me how this is clear? Or point me in the right direction?

## Answer by Quantuple (score 4)

https://quant.stackexchange.com/a/63382

Maybe this deck by Jim Gatheral would help get the intuition, see slides 10 and following.

The dynamics you mentioned is obtained by:

- Looking at the Bergomi dynamics for the forward variance process;

- Assuming there is only one factor driving the dynamics;

- Noticing a similarity with a rough process when replacing the Bergomi exponential kernel by a Riemann-Liouville fBm kernel

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