Good explanation - never thought of it that way. So basically, lognormal says that the underlying is just as likely to double as it is to be cut in half, and a long neutral straddle should gain the same amount either way. Since the absolute amount to the upside is doubled, I would have to have positive delta to begin with to make this possible, right?
No, it isn't as likely to double as to be cut in half. It's a cumulative effect - which is why the lognormal distribution is also called a cumulative normal distribution. Think of it this way - let's say the futures are at 10, and you're comparing a 19 call with a 1 put. To get to 1, it is statistically "harder" the further the futures drop. From 10 to 9 is a drop of 10%. But to drop another point from 9 to 8, the futures need to move 11%. And so on. To finally drop from 2 to 1 will require a move of 50%. As the futures rise, however, it gets statistically "easier and easier." To go from 10 to 11 is also a move of 10%, but from 11 to 12 requires a move of only 9%. To move from 18 to 19 will require a move of only 5.5%. In actuality, you can divide the move from 10 to 9 and from 10 to 11 into 10 moves, and show that it's really a bigger statistical move from 10 to 9 than from 10 to 11. And you can divide each of those steps into 10 parts. And on and on. That's why it's "cumulative." And that's why the ATM straddle has a positive delta, if you assume a lognormal distribution - which all models do. Because it's statistically a little easier to go up than down. Theoretically, the lognormal distribution is justified of course. Actual market prices of options reflect a more complex reality, and nobody has the final say as to what's "right."
That part in italics can't possibly be right. There's no way a straddle will do equally well whether the index moves 1400 points up or 700 points down. Even if the index is as likely to move up 1400 or down 700, the strike prices you're looking at are so much closer to the money than 700-1400 points that the lognormal aspect of the distribution almost doesn't matter. Put another way, a log curve "centered" at 1400 is almost perfectly linear over a +/- 25 point range. In other words, zero and infinity are both sufficiently far away that they don't really matter.
Lognormal distribution does not say that. Right idea, but not that extreme (the part in italics is completely wrong of course). Please see my previous post.