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Let's say we have option strike price equal to current stock price. And we have zero risk-free rate. In this case I assume that probability of exercise is 0.5 because chances that price will go up or down are equal.

As I understand $N(D_2)$ is exactly the probability of exercise and it should be 0.5 . It means that $D_2$ should be 0 in this case. But if we put equal current and strike price and zero rate in its formula it will not be zero:

$D_2=(Ln(S/K) + (r - 0.5*Vol^2)*t)/(Vol*Sqrt(t))$

Here $Ln(S/K)$ and $r$ become zero, but we still have $-0.5*Vol^2*t/(Vol*Sqrt(t))$ that is not zero.

Where is the mistake?

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expectations and medians are different things. If the strike is at the median, you get 0.5. However, the median of the log is lower and is

$$ \log S_0 - 0.5 \sigma^2 T $$ The median of $S_T$ is the exp of this.

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