  We present a Hawkes modeling of the volatility surface's high-frequency
dynamics and show how the Hawkes kernel coefficients govern the surface's skew
and convexity. We provide simple sufficient conditions on the coefficients to
ensure no-arbitrage opportunities of the surface. Moreover, these conditions
reduce the number of the kernel's parameters to estimate. Finally, we show that
at the macroscopic level, the surface is driven by a sum of risk factors whose
volatility processes are rough.
