According to most books the ATM option is the option with a delta of 0.50. However, this is only the case when the distribution is normal. The more positively skewed the distribution, the further the 0.50 delta option is out-of-the-money (for calls). According to the following article, the formula to calculate the 0.50 delta option strike is equal to:
S x e^(σ^2/2)
I want to know why this is exactly the case. Looking at the delta defintion I have:
delta = N(d1) = 0.50
Therefore,
d1 = 0
And
So, how do I get from this well-known formula to the above mentioned formula? Thanks in advance,