You can set a true roughness to build a market. But with real data, nobody hands you the answer — you have to estimate it back from what you observe. The catch: different estimators give different answers, and they disagree most exactly where markets live. Set a true H below, and watch three standard estimators try to recover it.
The green marker is the true roughness. At 0.5 (ordinary Brownian motion) all three estimators land near it — they agree. Roughness is identifiable here.
Watch the three coloured markers spread apart and drift off the true value — two bias up, one biases down. The rougher the truth, the worse they disagree.
In the rough regime the estimators can sit on both sides of Brownian — some say “rough”, some “smooth”. When they can't even agree on that, the roughness is not identifiable. That ambiguity is the finding.