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?