Replace Bergomi's exponential kernel with a Volterra power-law kernel and you get rough vol — smiles that match short-dated SPX out to the wings, term-structure of ATM skew that obeys the empirical $T^{H - 1/2}$ scaling.
Imagine a vol shock today that doesn’t fade cleanly — it echoes, with diminishing but long-lived force, into every future day. That’s a memory kernel: instead of vol reverting exponentially to a mean, past shocks pile up with a slowly-decaying weight. Traders see this in SPX data: after a selloff, 1-day implied vol skew is four or five times steeper than the 1-year skew, and the decay follows a power law rather than the flat term structure that Heston or standard Bergomi predict. Rough Bergomi (Bayer-Friz-Gatheral 2016) captures this by replacing the exponential variance kernel with a Volterra power-law kernel driven by fractional Brownian motion with Hurst exponent $H \approx 0.1$. The payoff is a closed-form ATM skew scaling law $\eta\,T^{H - 1/2}$ that explodes toward short maturities exactly as SPX data demands — a signature no classical model can replicate.
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Worked example
Under rough Bergomi with Hurst $H = 0.1$, the ATM skew of implied volatility scales as $T^{H - 1/2}$. What is the ratio of the ATM skew of a 1-week ($T_1 = 1/52$ year) option to that of a 1-year ($T_2 = 1$ year) option on the same name?
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Reflection
Classical stochastic-vol models (Heston, Bergomi) cannot reproduce the empirical $T^{-0.4}$ skew scaling without contradicting longer-dated fits. What does this mean structurally — what assumption is the rough-vol programme letting go of, and what price are practitioners paying for fitting the short end?