The other route to the roughness runs through option prices. Fit the rough-Heston model to a market implied-vol smile and read off its parameters — level ξ₀, skew ρ, vol-of-vol ν, and the roughness H. The model fits beautifully. But watch what H does when you slide it.
The dots are the market smile; the teal curve is the calibrated model. It sits right on the data — and it pins the level (ξ₀), the skew (ρ) and the vol-of-vol (ν) cleanly.
The fit doesn't budge. As you change H, the vol-of-vol ν silently recalibrates to compensate — the faint dashed line shows what H alone would do, and ν cancels it. Same smile, any H.
A single smile fixes ξ₀, ρ and ν, but leaves H on a flat, ν-degenerate direction — the calibration is ill-posed (cond ≈ 6×105). You can fit the market perfectly and still not know the roughness. Same wall as the price-history route.