No-arbitrage implies power-law market impact and rough volatility

TL;DR

This paper demonstrates that under no-arbitrage conditions, market impact follows a power-law, which naturally leads to rough volatility in prices, explaining universal market behaviors through mathematical analysis of stochastic equations.

Contribution

It establishes a theoretical link between no-arbitrage, power-law market impact, and rough volatility, supported by new mathematical results on stochastic Volterra equations.

Findings

Market impact must be power-law under no-arbitrage.

Power-law impact implies diffusive prices with rough volatility.

The impact exponent relates directly to the volatility's Hurst parameter.

Abstract

Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that under no-arbitrage assumption, the market impact function can only be of power-law type. Furthermore, we prove that this implies that the macroscopic price is diffusive with rough volatility, with a one-to-one correspondence between the exponent of the impact function and the Hurst parameter of the volatility. Hence we simply explain the universal rough behavior of the volatility as a consequence of the no-arbitrage property. From a mathematical viewpoint, our study relies in particular on new results about hyper-rough stochastic Volterra equations.