Gamma
Gamma is an estimate of how much the delta of an option changes when the price of the stock moves $1.00. As a tool, gamma can tell you how “stable” your delta is. A big gamma means that your delta can start changing dramatically for even a small move in the stock price.
Long calls and long puts both always have positive gamma. Short calls and short puts both always have negative gamma. Stock has zero gamma because its delta is always 1.00 – it never changes. Positive gamma means that the delta of long calls will become more positive and move toward +1.00 when the stock prices rises, and less positive and move toward 0.00 when the stock price falls. It means that the delta of long puts will become more negative and move toward –1.00 when the stock price falls, and less negative and move toward 0.00 when the stock price rises. The reverse is true for short gamma.
For example, the XYZ Aug 50 call has a delta of +.45, and the XYZ Aug 50 put has a delta of -.55, with the price of XYZ at $48.00. The gamma for both the XYZ Aug 50 call and put is .07. If XYZ moves up $1.00 to $49.00, the delta of the XYZ Aug 50 call becomes +.52 (+.45 + ($1 * .07), and the delta of the XYZ Aug 50 put becomes -.48 (-.55 + ($1 * .07). If XYZ drops $1.00 to $47.00, the delta of the XYZ Aug 50 call becomes +.38 (+.45 + (-$1 * .07), and the delta of the XYZ Aug 50 put becomes -.62 (-.55 + (-$1 * .07).
Position gamma measures how much the delta of a position changes when the stock price moves $1.00. Position gamma is calculated much in the same way as position delta.
Just as delta changes, so does gamma. If you were to look at a graph of gamma versus the strike prices of the options, it would look like a hill, the top of which is very near the ATM strike. Gamma is highest for ATM options, and is progressively lower as options are ITM and OTM. This means that the delta of ATM options changes the most when the stock price moves up or down. Let’s look at a deep ITM call option (delta near 1.00), an ATM call option (delta near .50), and an OTM call option (delta near .10). If the stock rises, the value of the ITM call will increase the most because it acts most like stock. Even though the ITM call has positive gamma, its delta really doesn’t get much closer to 1.00 than before the stock rose. The value of the OTM call will also increase, and its delta will probably increase as well, but it will still be a long way from 1.00. The value of the ATM option increases, and its delta changes the most. That is, its delta is moving closer to 1.00 much quicker than the delta of the OTM call. Practically speaking, the ATM call can provide a good balance of potential profit if the stock rises versus loss if the stock falls. The OTM call will not make as much money if the stock rises, and the ITM will lose more money if the stock falls.
Judging how gamma changes as time passes and volatility changes depends on whether the option is ITM, ATM or OTM. Time passing or a decrease in volatility acts as if it’s “pulling up” the top of the hill on the graph of gamma, and making the slope away from the top steeper. What happens is that the ATM gamma increases, but the ITM and OTM gamma decreases. The gamma of ATM options is higher when either volatility is lower or there are fewer days to expiration. But if an option is sufficiently OTM or ITM, the gamma is also lower when volatility is lower or there are fewer days to expiration.
What this all means to the option trader is that a position with positive gamma is relatively safe, that is, it will generate the deltas that benefit from an up or down move in the stock. But a position with negative gamma can be dangerous. It will generate deltas that will hurt you in an up or down move in the stock. But all positions that have negative gamma are not all dangerous. For example, a short straddle and a long ATM butterfly both have negative gamma. But the short straddle presents unlimited risk if the stock price moves up or down. The long ATM butterfly will lose money if the stock price moves up or down, but the losses are limited to the total cost of the butterfly.
Gamma is a good reason to look at a profit/loss graph of your position over a wide range of possible stock prices.
