In the Schweser it says " Hedges with at the money options will have higher gammas, and consequently small changes in stock price will lead to large changes in delta and frequent rebalancing"

This is saying that if the delta is higher, option prices will be very sensitive to stock prices, so there will have to be constant rebalancing. This makes sense.

However, what about the change of delta? When is the change of delta at it’s highest point. i would think that this point would be somewhere between where delta is very low and very high. There should be a turning point where delta hedges will need to be rebalanced a lot. If the change in delta (gamma) is at its highest (wherever that is), the difficult rebalancing begins.

the book says that gamma is at its largest when a call or put is at the money. If a call is at the money, shouldn’t delta be at its highest? but not necessarily the change in delta? I would think gamma would be higher before the strike price. where there is more upside movement of the price as it moves away from the “severely out of the money” to the at the money point. I would think that the change of delta when it is almost in the money would be smaller as the slope is almost one.

No . . . it’s saying that if the option is at the money (where delta is +0.5 (calls) or -0.5 (puts)), option deltas will change very quickly with respect to changes in the price of the underlying. The problem isn’t that the option price is sensitive to the change in the stock price; the problem is that the sensitivity of the option price changes rapidly : from sensitive to insensitive or vice-versa.

That’s exactly what gamma measures.

When the option is at the money: when delta = 0.5 for calls or -0.5 for puts.

You’re correct: it’s exactly dead centre: |delta| = 0.5.

Thanks Magician… what do you mean that the point at where the gamma is highest is at 0.5? Is that the option value? I’d like to understand this inutitively. I would think that right after the stock price is at the money, the delta is at its peak… so any movement from there would mean a decrease in gamma.

Gamma is highest when delta = 0.5 for calls or -0.5 for puts; those values occur at the money.

If you think of an option that is close to expiration, it’s easy to see: delta is (very close to) zero if the option is out of the money, and (very close to) +1 for calls and -1 for puts if the option is in the money; thus, when the option is at the money, the delta changes very quickly from (near) zero to (near) one (in absolute value). That very quick change is the high gamma.