Gamma: the curve

Gamma measures how fast delta changes when the stock moves $1.

If delta is the speedometer, gamma is the accelerator. It tells you how quickly the speed itself is picking up or dropping off.

Our $100 call has a delta of 0.53 and a gamma of 0.055. Let the stock rise $1 and delta climbs to 0.59. Another dollar takes it to 0.64. Each step up makes the call a little more sensitive than the last.

Why it is called the curve

In chapter 3, delta was the slope of the value curve. If that curve were a straight line, its slope would never change and gamma would be zero. A share of stock works that way. Its delta is always exactly 1.

An option’s value curve bends upward. So as the stock rises, the tangent tilts more steeply. Gamma measures that bend. You can also see it in the lower chart below, which plots delta itself. Gamma is the slope of that line.

Gamma near expiry

Now drag the days slider toward zero. At the strike, gamma grows as expiry gets closer. With 90 days left it is 0.032. With 7 days left it is 0.115. With 1 day left it is 0.305, about ten times the 90-day figure.

The reason is in the top chart. As expiry nears, the smooth value curve folds into the sharp corner of the payoff chart from chapter 2. At that corner, delta has to jump from near 0 to near 1 across a few dollars of stock price. A fast jump in delta is high gamma.

Away from the strike it goes the other way. One day before expiry, with the stock at $95, gamma is just 0.0002. That call is almost certain to expire worthless, and a small move won’t change that.

Who gains from gamma

If you buy calls or puts, gamma works for you. Your delta grows as the stock moves your way and shrinks as it moves against you. Gains speed up and losses slow down.

If you sell options, it works against you in the same way. This is why at-the-money options in their last few days can swing so hard in price.

That extra sensitivity has a price. The options with the most gamma also lose value fastest as each day passes. That cost is theta.