Option delta and gamma

So I’m a bit confused about something I read in the Wiley books regarding these two option Greeks. The material shows a graph of the delta-based value of a call option as the stock price goes up and down, with the value estimation error increasing at both of the extremes of the possible stock values (both positive and negative). This makes sense to me, since delta-prescribed values for the call options are linear and don’t account for curvature of actual option value (like effective duration with fixed-income.

But I get lost when they move to gamma. The gamma adjustment to valuing options accounts for curvature (like convexity in fixed-income), and helps the overall estimate of the option’s value. But the text also states that gamma is largest when the call is close to or at the money. In the delta illustration, when the stock prices moved a small amount around the exercise price, delta is still a good estimator of call value. This is the same area where the text states that gamma is at its highest?? From the sounds of it (and delta graph), gamma would be highest when the stock price is far from the exercise price (since it is at this point that delta is the poorest estimator of call value), and thus would need gamma to be large to correct its stand-alone estimate the most. Yet the text states that gamma is almost zero at these points (where the call is either deep in or out of the money). Totally lost here, so any clarification would be greatly appreciated.

It is easy to see why gamma has to be zero when the option (lets assume a call for simplicity) is either very out of the money or very in the money.

(1) If the call is very out of the money a small change in price of the underlying will not change the option value that much (informal: “if you own a call with strike 100 it barely matters if the underlying security trades at 0.01 or 0.02”). This means the delta is zero both before and after the price move and hence gamma ( = derivative of delta with respect to price of the underlying) is also zero.

(2) If the call is very in the money any change in the underlying will translate to changes in option value on an almost 1-to-1 basis (informal: “if you own a call with strike 100 and the price of the underlying changes from 999999 to 1000000 the value of your call will also go up by approximately 1”). This means delta is 1 before and after the change in price of the underlying security and hence gamma is zero.

Note: (1) and (2) are only approximations and work better the shorter the time to expiration and the larger the distance of the price of the underlying to the strike price.

It stands to reason that if delta changes from zero to one between very ouf of the money and very in the money (and gamma = 0 at both these extremes) that in between gamma can’t be zero everywhere. You can show that the maximum gamma occurs at the strike price by calculating the second derivative of the black and scholes formula with respect to the price of the underlying (which is the definition of gamma) and finding the maximum of it.