hard · Quantitative Finance qf-core

Using the standard short-maturity approximation linking local and implied volatility, what is the relationship between the local-vol skew and the implied-vol skew at the money?

1. The local-vol skew is approximately TWICE the implied-vol skew (the '2x slope rule'): partialσ_loc/partial K≈ 2,partialσ_imp/partial K at the ATM strike
2. The local-vol skew is numerically equal to the implied-vol skew at every strike, since local volatility is simply implied volatility re-expressed as a function of spot instead of strike
3. The local-vol skew is approximately HALF the size of the implied-vol skew in this ATM region, because implied volatility represents the spatial average of local variance taken along the strike axis
4. The local-vol skew carries the opposite sign to the implied-vol skew across the whole surface, since local volatility behaves as the dual surface of implied volatility under a strike-spot reflection map

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