Rough volatility: from empirical evidence to the Rbergomi model

Abstract

The Black–Scholes–Merton model assumes constant volatility, contradicting the implied volatility surface of equity markets. A key feature of this surface is the at-the-money skew, which empirically explodes as a power law $\Psi(\tau) \sim \tau^{-0.4}$ as the maturity $\tau \to 0$. Classical stochastic volatility models such as Heston and Bergomi cannot reproduce this: driven by Brownian motion, their short-maturity skew stays bounded. Following Gatheral, Jaisson and Rosenbaum, we show that modelling log-volatility as a fractional Brownian motion with Hurst exponent $H \approx 0.1$ resolves this, and we develop the rough Bergomi model, whose Volterra kernel $(t-s)^{H-1/2}$ generates the empirical scaling $\Psi(\tau) \sim \tau^{H-1/2}$. We add three numerical contributions: a replication of the Hurst estimation on eight equity indices, a Monte Carlo validation of the method, and a Bergomi vs. rBergomi simulation comparison.