Quote from dmo:
There are two factors that make the ATM straddle have a positive delta. First, as Walter says, is the effect of the lognormal distribution. That is true even if the cost of carry is zero, as is true of options on futures.
The cost of carry effect - if there is a cost of carry - increases the delta of ATM straddles. As you say, a good way of thinking of that effect is that it raises the price of the underlying - so the ATM straddle isn't really ATM.
But even without any cost of carry, a true ATM straddle has a positive delta due to the lognormal distribution.
Quote from steve3052:
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?
Quote from steve3052:
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.
Quote from dmo:
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.
Quote from dmo:
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."