We report on simultaneous sub-second optical and X-ray timing observations of the low-mass X-ray binary black hole candidate MAXI J1820+070. The bright 2018 outburst rise allowed simultaneous photometry in five optical bands (ugrizs) with HiPERCAM/GTC (Optical) at frame rates over 100 Hz, together with NICER/ISS observations (X-rays). Intense (factor of 2) red flaring activity in the optical is seen over a broad range of time-scales down to similar to 10 ms. Cross-correlating the bands reveals a prominent anticorrelation on time-scales of similar to seconds, and a narrow sub-second correlation at a lag of approximate to+165 ms (optical lagging X-rays). This lag increases with optical wavelength, and is approximately constant over Fourier frequencies of similar to 0.3-10 Hz. These features are consistent with an origin in the inner accretion flow and jet base within similar to 5000 Gravitational radii. An additional similar to+5 s lag feature may be ascribable to disc reprocessing. MAXI J1820+070 is the third black hole transient to display a clear similar to 0.1 s optical lag, which may be common feature in such objects. The sub-second lag variation with wavelength is novel, and may allow constraints on internal shock jet stratification models.