## Understanding the “Greeks,” Part 2: Gamma

The analysis of options includes four important risk metrics, which are identified by the Greek letters delta, gamma, vega, and theta. (Based on that, it should not be difficult to understand why these four metrics are collectively nicknamed the “Greeks.”) Here we’ll examine what the second Greek, gamma, measures. (This discussion will require an understanding of the fundamental workings of options, so if you do not have a good grasp of those concepts, please first review some of our material on options basics.)

It is important to recognize that all of the Greeks are derived from mathematical models. They are not “hard” numbers like an option’s price, bid/ask spread, volume, or open interest. For that reason, the Greek measures are only as good as the model that underlies them—something you should carefully consider before relying heavily on a particular set of Greeks to develop a trading strategy. This is also why you will see different conventions for representing the value of any given Greek.

Gamma is directly related to delta, and in fact you may sometimes see it called the “first derivative of delta.” It measures the rate of change in delta per point of movement in the underlying stock price. Always a positive number, gamma is highest for options that are at-the-money or near-the-money, decreasing the more the option gets into-the-money or out-of-the-money. In short, gamma would graph as a bell curve.

Let’s look at an example. Bank of America (BAC) was trading at $8.02 on February 20, having closed down $0.07 or 0.87% in the last session. The chart below shows a selection of March options with the delta and gamma values for each as of February 20.

BAC March Call |
||

Strike Price |
Delta |
Gamma |

11.00 | 0.02 | 0.05 |

10.00 | 0.07 | 0.12 |

9.00 | 0.16 | 0.28 |

8.00 | 0.54 | 0.44 |

7.00 | 0.89 | 0.20 |

6.00 | 1.00 | 0.00 |

It’s important to remember the rationale for delta values. (You can review Part 1 of this series, which addresses delta, if need be.) Delta measures the relationship between changes in the option price and changes in the underlying stock price, with high delta values reflecting a nearly one-to-one linkage and a very low delta value reflecting almost no effect on the option price from changes in the underlying stock.

You’ll notice in the example that the $6 strike option has a delta of 1.00. This shows that—as of February 20, at least—the delta model sees no chance that BAC will drop to $6.00 before the March contract expires. The zero gamma indicates that this position is solid, since delta is not changing at all.

On the opposite end of the spectrum, the $11 strike option has a delta of only 0.02, which means at this point there is little expectation that BAC will reach that price prior to expiration. On the other hand, the 0.05 gamma means delta has fluctuated a bit, so there is some—albeit very little—uncertainty about the outcome.

At-the-money and near-the-money options are where the real uncertainty comes into play. This is only common sense, as the stock is far more likely to vary by $0.02, $0.05, or $0.10 in either direction within the time remaining than to vary by $2 or $3. Also, just as delta rises for at- or near-the-money options as expiration approaches, gamma also generally rises, and can in fact become quite large in the last few days before expiration.

Certain advanced option strategies utilize gamma, but in simple terms it can serve as an indication of trader sentiment regarding the expected price movement of a given stock, especially when examined in conjunction with delta.

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