Rough volatility models — where the volatility process has Hurst parameter H < 1/2, making it rougher than Brownian motion — fit implied volatility surfaces remarkably well. But the roughness seems arbitrary: why should volatility be rough?
The paper derives roughness from order flow. In an order-driven market, buy and sell orders arrive as a Poisson process. Each order has a persistent impact on volatility — not permanent (which would produce smooth volatility) and not instantaneous (which would produce white noise) but slowly decaying. The aggregate effect of many persistent order impacts, each small, converges to the rough Bergomi model.
The convergence is weak convergence of càdlàg processes — the discrete Poisson arrivals become the continuous rough process in the limit. The Clark-Ocone formula, adapted from Brownian to Poisson settings, provides the error rates: how fast the discrete microstructure model approaches the continuous rough limit.
The microstructural interpretation resolves the arbitrariness. The roughness parameter H is determined by the decay rate of order impact — how quickly the market absorbs each order's influence on volatility. Faster decay means smoother volatility; slower decay means rougher. The observed H ≈ 0.1 in equity markets corresponds to a specific, measurable rate of impact persistence.
From Poisson arrivals to fractional Brownian motion via the aggregation of persistent impacts. The roughness isn't a model choice — it's an emergent consequence of how orders accumulate in the book.