Long, long ago we learned about velocity and acceleration. Velocity told us the speed with which something was moving as well as the direction. Acceleration told us how fast the velocity was changing with relationship to time, as well as the direction it was changing (increasing the velocity or slowing it down).

Think of delta and gamma like that. Delta tells us how much our option trades will benefit or be hurt by a change in price movement. Gamma tells us how much delta will change with respect to price movement and in what direction. It's the acceleration. Let's look at some examples. Those who would like a quick primer on the basics of the Greek called "delta" can find them at this recent Options 101 article.

The first two things that we should know is that gamma is always positive for long options and negative for sold options, and that its absolute value is highest for at-the-money strikes. Therefore, if you buy an at-the-money call with a gamma of +3.00, you know that the gamma will grow smaller as you move away from the at-the-money strike, no matter which direction. Therefore, if the underlying takes off to the upside, the positive delta of your long call will grow, but it will grow by a smaller and smaller amount as the underlying climbs because that gamma shrinks the further away from the money you get.

For example, on October 4, 2013, at 12:55 pm CT, delta reduces by 7 points between an ATM 1080 RUT NOV 13 call with a delta of 51 and a 1090 one with a delta of 44. It changes by only 4 points, however, between an in-the-money 1030 RUT NOV 13 call with a delta of 76 and a 1020 one with a delta of 80. The differences between the deltas of the different strikes are growing smaller, a direct effect of the smaller gamma gets the further OTM a strike is.

If price drops, your profit-and-loss line on a long call is hurt, of course, due to the delta effect. Delta is also reduced, however, by the amount of gamma. That means that the next one-point drop will also hurt the profit-and-loss line, but it will hurt by a smaller amount because the positive delta isn't quite as large. Eventually, if price drops enough, the gamma grows so small that it doesn't make a measurable difference in delta any longer.

Also, remember that a long put has a negative delta but a positive gamma. Thinking about gamma's effect on delta gets a little more complicated in this instance. If price rises, we know that that the profit-and-loss line theoretically dips by the amount of the delta. However, delta grows less negative as prices climb. Put another way, the delta is still negative but the absolute value of the delta shrinks.

At 12:59 pm CT on October 4, 2013, an ATM 1080 RUT NOV 13 long put had a delta of -49, while a 1020 RUT one's delta was -22. A long put's profit-and-loss line will always be hurt when the underlying climbs, of course, but the positive gamma means the loss per one-point increment grows smaller as the underlying climbs, as long as other factors such as the volatility and time to expiration remain constant. Looking at it another way that may make you wince but helps you remember the effect. After you've lost almost all the value in your put, another 1-point climb in the underlying isn't going to make your loss that much bigger than it was one point lower down.

What about a position with negative gammas?

At-the-Money Call Butterfly, Established 31 DTE, OptionOne Explorer Chart:

This trade was simulated using actual mid prices and included commissions of \$1.25/contract but no built-in slippage. Delta measures -27.09, meaning that over a one-point change in the RUT's price, the profit theoretically changes +/- \$27.09 if all other factors stay the same. The negative delta means that if the RUT rises a point, the loss increases by approximately \$27.09 if none of the other inputs change. If the RUT drops a point, the current \$50.00 loss is reduced by \$27.09, if no other factors change.

However, as the underlying's price moves, delta changes, and gamma gives us an idea of how much it changes and in which direction. In this butterfly shown above gamma was -3.42. Figuring out how the delta will change will take some bending-of-the-mind for those who are not mathematically inclined, but there's an easy way to remember the impact. Keep in mind a truism that helps: negative gamma is bad on the way up and bad on the way down. I believe I first heard that truism in a CBOE webinar by former market maker Dan Sheridan, but it may have been CBOE instructor James Bittman, author of Trading Options as a Professional who said it. Whoever said it, it's a great truism to help us untangle the effect of a negative gamma.

If we know that a negative gamma hurts on the way down as well as on the way up, that means that the move is going to change delta in a way that hurts our trade, no matter which direction the underlying goes. The absolute value of the gamma (3.42) tells us how much it will be hurt, and we can intuit that the original negative delta will be less negative as the RUT's price heads lower by a point, and it will be less negative by 3.42 deltas if no other inputs into the pricing of options change. The negative gamma reduces the amount we would gain for a price movement in the "right" direction: hence, the "bad" effect on the way down.

If we know that a negative gamma hurts on the way up, too, we know that delta will become more negative on the way up, and it will do so by approximately -3.42 deltas, if no other inputs into the pricing of options changes. Therefore, after a one-point rise in the RUT's price, the delta would be approximately -27.09 - 3.42 = -30.51.

When that first chart was snapped, the RUT's price was 1038.08. Let's check that theory about negative gamma.

Same Position, Same Day, Price at 1039.44:

Price is now 1039.44, so 1.36 points above the original price. Delta is more negative than the original -27.09, and also more negative than the -30.51 that we anticipated it would be from only a 1-point rise rather than a 1.36-point rise. This fits with what that truism tells us about negative gamma. If gamma was originally -3.42, we can roughly approximate that a rise of 1.36 points would have resulted in a 1.36(-3.42) = -4.65 more negative delta. The new delta might be approximated at -27.09 - 4.65 = -31.74. We can see that the actual -32.22 delta wasn't so far off. We can presume that implied volatility was also changing. Therefore, a negative gamma means that the profit-and-loss is negatively impacted both on the way up and the way down.

However, when you're selling more premium than you are buying, as you do with at-the-money butterflies or iron condors centered around the price at entry, you're going to have that negative gamma.

Another truism is that gamma changes with time.

Same Position, Thirteen Days Later, Price at 1039.36:

With the RUT price just four cents different than the last chart, and with implied volatilities very near what was present on the first chart, we can isolate the effect of the passage of time on gamma. Gamma is now -7.63. The absolute value has increased, so a move away from the current price is going to adversely impact delta more than it would have earlier in the trade.

Who cares, you might be thinking as you look at that chart. Profit is 26 percent of the margin in the trade. Many traders would have long-ago locked in the profit gifted by prices churning in a tight range. However, sometimes traders are battling that gamma effect when they already have a loss on the trade, or they're battling it closer to expiration, when it can make one's profit and loss quite volatile.

Gamma is a much more complex and interesting Greek than has been explained here, but this is a basic article and not a book. Bittman's book includes a section on gamma, and a search of educational articles on CBOE and the educational arm of the OIC, found at optionseducation.org might turn up helpful information, too.

Linda Piazza