Excel Models
The Greeks – Gamma (Γ)
Also, the relationship between the option price and the underlying price is not linear. Gamma determines the degree of curvature of this relationship. It is the second derivative that affects the option price’s sensitivity when the stock price changes over a wide range. (For small changes, delta is sufficient, because the curvature does not have a significant effect.)
The greater the curvature, meaning the higher the Gamma, the greater the impact the stock price has on the option price, and therefore the more important it is to take gamma into account when hedging risk.
It is important to understand that the gamma of the stock itself is zero, meaning that adding or removing shares does not affect the portfolio’s gamma. In other words, a portfolio’s gamma is the sum of the gammas of the options, so it can be neutralized using options.
If we look at a single asset and a single option, we will see that the gamma of short options can be neutralized by buying the same options, which is pointless in practice; however, the approach becomes meaningful when we are talking about a portfolio of different options written on the same underlying stock. The gamma of such a portfolio, just like delta, is additive.
Options on the same underlying may differ by strike, maturity, type, and quantity; the portfolio may also include futures.
Moreover, when an option is added to a portfolio to make it gamma-neutral, the portfolio’s delta changes (since delta is additive).
Therefore, for the portfolio to return to a delta-neutral position, a corresponding change must be made by buying or selling the underlying asset/stock.
Gamma is calculated using the formula:
N′(d1) – the normal probability density function (in Excel – The Greeks).
An option’s gamma is high when the option price is close to the strike price.
Gamma also decreases proportionally as the option approaches expiration, although along different trajectories.
Source:
Options, Futures & Other Derivatives, John C. Hull