Consider the Taylor expansion of the integrand of the cumulative normal function for x ≈ 0, i.e.

Thereby, find an approximation for the price of a lognormal option close-to-the-money. Attempt something similar for a normal option by doing a Taylor expansion for  in addition. Hence obtain a normal vol in terms of a lognormal vol, by neglecting terms higher than O(x) in the expansions for N(x) and n(x).

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N(x): = 0.5 + /2 S e- dz 27 = 0.5 + 2 S (1 1 2 + 3 z + .. )dz.