hard · Quantitative Finance

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 equals the implied-vol skew, since local vol is just implied vol expressed as a function of spot rather than strike
3. The local-vol skew is approximately HALF the implied-vol skew, because implied vol is the spatial average of local variance along the strike
4. The local-vol skew has the opposite sign to the implied-vol skew, since local vol is the dual of implied vol under the strike-spot reflection

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