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What is the reason that At the Money options have the highest Gamma? I understand that Gamma is the rate of change of an options price w.r.t the change of the underlying price (a.ka. the change of delta).
So for deep in the money option (which has an almost constant delta of 1) of deep out of the money option (which has an almost constant delta of +1 or -1), gamma is 0 since delta is constant. Is this correct? But how does that imply that Gamma is highest for at the money option?
Yes, you've got it right. The change in price is less meaningful as the instrument is further from the price of the underlying. As the delta moves less, the gamma is much less.
Gamma is to delta as acceleration is to speed. Speed is movement relative to X, and acceleration is rate of change in speed. Delta is movement relative to S, and gamma is the rate of change in delta.
Delta changes quickly when it is around the money, which is another way of saying gamma is higher. Delta is the change of the option price relative to the change in stock price.
If the strike price is near the market price, then the odds of being in or out of the money could appear to be changing very quickly - even going back and forth repeatedly.
Gamma is the rate of change of the delta, so these sudden lurches in pricing are by definition the gamma.
This is to some extent a little mundane and even obvious. But it's a useful heuristic for analyzing prices and movement, as well as for focusing analyst attention on different pricing aspects.
You've got it right. If delta is constant (zero 'speed' for the change in price) then gamma is zero (zero 'acceleration').
