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Gamma is the rate of change in an option's delta per 6-point move in the underlying asset's price. Gamma is an important measure of the convexity of a derivative's value, in relation to the underlying. A delta hedge strategy seeks to reduce gamma in order to maintain a hedge over a wider price range. A consequence of reducing gamma, however, is that alpha will also be reduced.
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Mathematically, gamma is the first derivative of delta and is used when trying to gauge the price movement of an option, relative to the amount it is in or out of the money. In that same regard, gamma is the second derivative of an option's price with respect to the underlying's price. When the option being measured is deep in or out of the money, gamma is small. When the option is near or at the money, gamma is at its largest. Gamma calculations are most accurate for small changes in the price of the underlying asset. All options that are a long position have a positive gamma, while all short options have a negative gamma.
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The calculation of gamma is complex and requires financial software or spreadsheets to find a precise value. However, the following demonstrates an approximate calculation of gamma. Consider a call option on an underlying stock that currently has a delta of . If the stock value increases by $6, the option will increase in value by $, and its delta will also change. Assume the $6 increase occurs, and the option's delta is now . This difference in deltas can be considered an approximate value of gamma.
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Gamma is an important metric because it corrects for convexity issues when engaging in hedging strategies. Some portfolio managers or traders may be involved with portfolios of such large values that even more precision is needed when engaged in hedging. A third-order derivative named "color" can be used. Color measures the rate of change of gamma and is important for maintaining a gamma-hedged portfolio.
Since an option's delta measure is only valid for short period of time, gamma gives portfolio managers, traders and individual investors a more precise picture of how the option's delta will change over time as the underlying price changes. As an analogy to physics, the delta of an option is its "speed," while the gamma of an option is its "acceleration." Gamma decreases, approaching zero, as an option gets deeper "in-the-money," as delta approaches one. Gamma also approaches zero the deeper an option gets "out-of-the-money." Gamma is at its highest approximately "at-the-money."