Gamma is a measure of risk for options, and reflects the rate of change in delta for a $1 change in the underlying asset price. In other words, gamma represents what happens to delta as the underlying asset moves. Because it represents the change in delta as the underlying price changes, Gamma can be used to determine the magnitude of movement in an options price for a corresponding movement in the underlying. Gamma is an essential part of derivatives pricing, and is very useful when it comes to more complicated strategies of risk management.
Gamma is always positive for long options and negative for short options. This means all owners of options have positive gamma, regardless of being long or short the underlying asset. This also means that long option buyers benefit the most from gamma, as for every $1 movement in favor of the position, like a $1 increase for calls or a $1 decrease for puts, the opportunity for profit increases. This is because as gamma increases, so does delta, and meaning the profit per dollar increases as well. Furthermore, if the underlying asset moves against a position, like decreasing in price for calls or increasing in price for puts, gamma causes delta to decrease, causing lesser losses.
Additionally, like theta and delta, gamma varies based on an options strike price and expiration. Gamma increases as an option approaches expiration, and decreases as an option becomes deeper in or out the money. The closer the strike price is to the underlying asset price, the higher the gamma, meaning at the money options have the highest gamma. Gamma is highest when the option is at the money, and decreases as the asset price moves away from the strike price.
How is Gamma used?
Gammas mainly used to analyze delta, as it shows how delta will change with different movements in the underlying asset price. This, in turn, allows for risk mitigation, as an options price can be analyzed for a variety of different situations.
In other words, by looking at gamma, you can tell how much delta will change when the underlying asset’s price changes, which allows for a rough estimate of an options price for different movements in the underlying.
Additionally, gamma can be used in more advanced methods of portfolio management. When delta hedging, gamma must be taken into account due to its impact on delta. As a quick reminder, delta hedging is a technique that minimizes delta so underlying price movement has little effect on a position. Delta neutral positions utilize this concept of delta hedging. Delta neutral positions have a delta close to 0 so underlying asset movement causes little change in the positions. This requires a neutral or near 0 gamma, which ensures the delta hedge remains effective even with more drastic price movement in the underlying asset.
An example with Gamma
Let's look at an example of gamma and its relation to delta. If a call option has a delta of 0.25, and a gamma of 0.05, a $1 change in the underlying asset price will result in delta changing by 0.05 (which is the gamma value). As always, this is assuming no other factors change. If the underlying asset’s price increased by $1, the delta would go up to 0.30. Similarly, if the underlying asset price decreased by $1, the delta would go down to 0.20. As we discussed before, gamma is always a positive value. This can be confusing when it comes to options like long puts, so let's look at another example. If a long put has a delta of -0.25 and a gamma of 0.05, then a $1 increase in underlying asset would cause the delta to go to -0.20, while a $1 decrease in underlying asset price would result in delta going to -0.3.
The math behind Gamma
We know delta to be the first derivative of the option value V with respect to the underlying asset price S. Additionally, we know that gamma represents the rate of change of delta with respect to changes in the underlying asset price. This means that gamma is the second derivative of the option value with respect to the underlying asset, and the first derivative of delta with respect to the underlying asset price.
Wider applications of Gamma
Gamma helps represent the concept of convexity, which states that as the price of an underlying asset changes, the price of a derivative (like an option), does not change linearly. Instead, its change depends on the second derivative.
In mathematical terms, convexity represents the second derivative of the output price with respect to an input price, as does gamma in regard to derivative pricing. Convexity is most commonly used in reference to bond pricing and bond yields.
Convexity = 1 / [bond price x (1+y)2]Σ
Gamma represents the relationship between underlying asset price and delta.
Gamma varies depending on an options strike price and expiration, with at the money options and options closest to expiration having the highest gamma.
All long options have positive gamma, and all short options have negative gamma.
Gamma can be used to measure the magnitude of price movement for an option when the underlying asset changes, due to its relationship with delta.