Demonstrate, that for a volatility surface b(K, T ) which is nonarbitrageable everywhere according to a proportional

Demonstrate, that for a volatility surface bσ(K, T ) which is nonarbitrageable everywhere according to a proportional dividend model, that the ‘sticky delta’ transformation:

under spot shock S = S′−S will preserve the no arbitrage conditions for arbitrary spot shock. (Hint: consider the Dupire local vol formula.)

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'(K,T)=(KS/S',T)