by John Summa (Contact Author | Biography)
Gamma is one of the more obscure Greeks. Delta, Vega and Theta generally get most of the attention, but Gamma has important implications for risk in options strategies that can easily be demonstrated. First, though, let's quickly review what Gamma represents.
As was presented in summary form in Part II of this tutorial, Gamma measures the rate of change of Delta. Delta tells us how much an option price will change given a one-point move of the underlying. But since Delta is not fixed and will increase or decrease at different rates, it needs its own measure, which is Gamma.
Delta, recall, is a measure of directional risk faced by any option strategy. When you incorporate a Gamma risk analysis into your trading, however, you learn that two Deltas of equal size may not be equal in outcome. The Delta with the higher Gamma will have a higher risk (and potential reward, of course) because given an unfavorable move of the underlying, the Delta with the higher Gamma will exhibit a larger adverse change. Figure 9 reveals that the highest Gammas are always found on at-the-money options, with the January 110 call showing a Gamma of 5.58, the highest in the entire matrix. The same can be seen for the 110 puts. The risk/reward resulting from changes in Delta are highest at this point. (For more insight, see Gamma-Delta Neutral Option Spreads.)
In terms of position Gamma, a seller of put options would face a negative Gamma (all selling strategies have negative Gammas) and buyer of puts would acquire a positive Gamma (all buying strategies have positive Gammas. But all Gamma values are positive because the values change in the same direction as Delta (i.e., a higher Gamma means a higher change in Delta and vice versa). Signs change with positions or strategies because higher Gammas mean greater potential loss for sellers and, for buyers, greater potential gain.
Gammas along a strike chain reveal how the Gamma values change. Take a look at Figure 9, which again contains an IBM options Gamma matrix for the months of January, February, April and July. If we take the out-of-the-money calls (indicated with arrows), you can see that the Gamma rises from 0.73 in January for the 125 out-of-the-money calls to 5.58 for January 115 at-the-money calls, and from 0.83 for the out-of-the-money 95 puts to 5.58 for the at-the-money 110 puts.
Perhaps more interesting, however, is what happens to Delta and Gamma values across time when the options are out-of-the-money. Looking at the 115 strikes, you can see in Figure 11 that the Gammas rise from 1.89 in July to 4.74 in January. While lower levels than for the at-the-money call options (again always the highest Gamma strike whether puts or calls), they are associated with falling, not rising Delta values, as seen in Figure 10. While not circled, the 115 calls show Deltas for July at 47.0 and 26.6 for January, compared with a drop from 56.2 in July to just 52.9 in January for the at-the-money Deltas. This tells us that the while out-of-the-money January 115 calls have gained Gamma, they have lost significant Delta traction from time value decay (Theta).
What do the Gamma values represent?
A Gamma of 5.58 means that for each one-point move of the underlying, Delta on that option will change by +5.58 (other things remaining the same). Looking at the Delta for the 105 January puts in Figure 10 for a moment, which is 23.4, if a trader buys the put, he or she will see the negative Delta on that option increase by 3.96 Gammas x 5, or by 19.8 Deltas. To verify this, take a look at the Delta value for the at-the-money 110 strikes (five points higher). Delta is 47.1, so it is 23.7 Deltas higher. What accounts for the difference? Another measure of risk is known as the Gamma of the Gamma. Note that Gamma is increasing as the put moves closer to being at-the-money. If we take an average of the two Gammas (105 and 110 strike Gammas), then we will get a closer match in our calculation. For example, the average Gamma of the two strikes is 4.77. Using this average number, when multiplied by 5 points, gives us 22.75, now only one Delta (out of 100 possible Deltas) shy of the existing Delta on the 110 strike of 23.4. This simulation helps to illustrate the dynamics of risk/reward posed by how rapidly Delta can change, which is linked to the size and rate of change of Gamma (the Gamma of the Gamma).
Finally, when looking at Gamma values for popular strategies, categorization, much like with position Theta, is easy to do. All net selling strategies will have negative position Gamma and net buying strategies will have net positive Gamma. For example, a short call seller would face negative position Gamma. Clearly, the highest risk for the call seller would be at-the-money, where Gamma is highest. Delta will increase rapidly with an adverse move and with it unrealized losses. For the buyer of the call, it is where potential unrealized gains are highest for a favorable move of the underlying.
Conclusion
Gamma tells us how fast Delta changes when the underlying moves, but it has characteristics that are not so obvious across time and vertically along strike chains for different months. Some patterns in Gammas inside this matrix of strike prices were highlighted, with an interpretation of the risk/reward significance. Finally, position Gammas for popular strategies is presented in table format.
Gamma is one of the more obscure Greeks. Delta, Vega and Theta generally get most of the attention, but Gamma has important implications for risk in options strategies that can easily be demonstrated. First, though, let's quickly review what Gamma represents.
As was presented in summary form in Part II of this tutorial, Gamma measures the rate of change of Delta. Delta tells us how much an option price will change given a one-point move of the underlying. But since Delta is not fixed and will increase or decrease at different rates, it needs its own measure, which is Gamma.
Delta, recall, is a measure of directional risk faced by any option strategy. When you incorporate a Gamma risk analysis into your trading, however, you learn that two Deltas of equal size may not be equal in outcome. The Delta with the higher Gamma will have a higher risk (and potential reward, of course) because given an unfavorable move of the underlying, the Delta with the higher Gamma will exhibit a larger adverse change. Figure 9 reveals that the highest Gammas are always found on at-the-money options, with the January 110 call showing a Gamma of 5.58, the highest in the entire matrix. The same can be seen for the 110 puts. The risk/reward resulting from changes in Delta are highest at this point. (For more insight, see Gamma-Delta Neutral Option Spreads.)
Figure 9: IBM options Gamma values. Values taken on Dec. 29, 2007. The highest Gamma values are always found on the at-the-money options that are nearest to expiration. |
Source: OptionsVue 5 Options Analysis Software |
In terms of position Gamma, a seller of put options would face a negative Gamma (all selling strategies have negative Gammas) and buyer of puts would acquire a positive Gamma (all buying strategies have positive Gammas. But all Gamma values are positive because the values change in the same direction as Delta (i.e., a higher Gamma means a higher change in Delta and vice versa). Signs change with positions or strategies because higher Gammas mean greater potential loss for sellers and, for buyers, greater potential gain.
Gammas along a strike chain reveal how the Gamma values change. Take a look at Figure 9, which again contains an IBM options Gamma matrix for the months of January, February, April and July. If we take the out-of-the-money calls (indicated with arrows), you can see that the Gamma rises from 0.73 in January for the 125 out-of-the-money calls to 5.58 for January 115 at-the-money calls, and from 0.83 for the out-of-the-money 95 puts to 5.58 for the at-the-money 110 puts.
Figure 10: IBM options Delta values. Values taken on Dec. 29, 2007. |
Source: OptionVue 5 Options Analysis Software |
Figure 11: IBM Options Gamma values. Values taken on Dec. 29, 2007. |
Source: OptionVue 5 Options Analysis Software |
Perhaps more interesting, however, is what happens to Delta and Gamma values across time when the options are out-of-the-money. Looking at the 115 strikes, you can see in Figure 11 that the Gammas rise from 1.89 in July to 4.74 in January. While lower levels than for the at-the-money call options (again always the highest Gamma strike whether puts or calls), they are associated with falling, not rising Delta values, as seen in Figure 10. While not circled, the 115 calls show Deltas for July at 47.0 and 26.6 for January, compared with a drop from 56.2 in July to just 52.9 in January for the at-the-money Deltas. This tells us that the while out-of-the-money January 115 calls have gained Gamma, they have lost significant Delta traction from time value decay (Theta).
What do the Gamma values represent?
A Gamma of 5.58 means that for each one-point move of the underlying, Delta on that option will change by +5.58 (other things remaining the same). Looking at the Delta for the 105 January puts in Figure 10 for a moment, which is 23.4, if a trader buys the put, he or she will see the negative Delta on that option increase by 3.96 Gammas x 5, or by 19.8 Deltas. To verify this, take a look at the Delta value for the at-the-money 110 strikes (five points higher). Delta is 47.1, so it is 23.7 Deltas higher. What accounts for the difference? Another measure of risk is known as the Gamma of the Gamma. Note that Gamma is increasing as the put moves closer to being at-the-money. If we take an average of the two Gammas (105 and 110 strike Gammas), then we will get a closer match in our calculation. For example, the average Gamma of the two strikes is 4.77. Using this average number, when multiplied by 5 points, gives us 22.75, now only one Delta (out of 100 possible Deltas) shy of the existing Delta on the 110 strike of 23.4. This simulation helps to illustrate the dynamics of risk/reward posed by how rapidly Delta can change, which is linked to the size and rate of change of Gamma (the Gamma of the Gamma).
Finally, when looking at Gamma values for popular strategies, categorization, much like with position Theta, is easy to do. All net selling strategies will have negative position Gamma and net buying strategies will have net positive Gamma. For example, a short call seller would face negative position Gamma. Clearly, the highest risk for the call seller would be at-the-money, where Gamma is highest. Delta will increase rapidly with an adverse move and with it unrealized losses. For the buyer of the call, it is where potential unrealized gains are highest for a favorable move of the underlying.
Strategies | Position Gamma Signs |
Long Call | Positive |
Short Call | Negative |
Long Put | Positive |
Short Put | Negative |
Long Straddle | Positive |
Short Straddle | Negative |
Long Strangle | Positive |
Short Strangle | Negative |
Put Credit Spread | Negative |
Put Debit Spread | Positive |
Call Credit Spread | Negative |
Call Debit Spread | Positive |
Call Ratio Spread | Negative |
Put Ratio Spread | Negative |
Put Back Spread | Positive |
Call Back Spread | Positive |
Calendar Spread | Positive |
Covered Call Write | Positive |
Covered Put Write | Positive |
Figure 12: Position Gamma signs for common strategies for options. The position Gammas in this table represent standard strategy setups. |
Conclusion
Gamma tells us how fast Delta changes when the underlying moves, but it has characteristics that are not so obvious across time and vertically along strike chains for different months. Some patterns in Gammas inside this matrix of strike prices were highlighted, with an interpretation of the risk/reward significance. Finally, position Gammas for popular strategies is presented in table format.
Next: Options Greeks: Position Greeks »
Table of Contents
- Options Greeks: Introduction
- Options Greeks: Options and Risk Parameters
- Options Greeks: Delta Risk and Reward
- Options Greeks: Vega Risk and Reward
- Options Greeks: Theta Risk and Reward
- Options Greeks: Gamma Risk and Reward
- Options Greeks: Position Greeks
- Options Greeks: Inter-Greeks Behavior
- Options Greeks: Conclusion
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