  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.
