Greek letters : GAMMA
The gamma represents the speed of adjustment of the Delta. Mathematically, it is the derivative of the Delta, and therefore the second derivative of the premium of the option compared to the price of the underlying S. The gamma of an option will be positive when the option is purchased, whether it is a call or a put. When selling Vanilla options, the Gamma will always be negative.
Here, N'(d1) is the density function of the reduced (0, 1) centered normal distribution, « e » the exponential, q represents the dividends, T the time, S0 the spot price of the underlying and σ the volatility. A low Gamma will result in a Delta that varies slowly, while a very high Gamma in absolute value will indicate that the Delta is very sensitive to changes in the price of the underlying. If this high Gamma is positive, the Delta will increase when the price of the underlying rises, and in the case of a negative Gamma, the Delta will decrease when the price of the underlying rises. The Gamma will always have the highest value in absolute terms when the option strike is at the price At The Money Forward (ATMF). The further the price of the chosen strike moves away from this price, the lower the Gamma will be.
As we can see, the Gamma evolves in a different way according to the chosen maturity of our option: the longer the maturity, the flatter the curvature of the Gamma will be (higher value at the wings and lower at delta 50, ATMF price).
On the contrary, the shorter the maturity will be, the higher the Gamma will be at ATMF price (and the lower it will be at the wings). This explains the fact that the Delta is more sensitive for a shorter maturity: because the high Gamma gives it a stronger acceleration. It is common to say that the shorter the maturity, the more Gamma the option
has in it. The sensitivity of the Gamma with respect to expiration time can be measured by the Greek letter « Color », obtained by deriving the Gamma function from the maturity T. It is necessary to specify that the Gamma accelerates especially after the Delta 25 and before the Delta 75. We will not take into account the Delta 75, as these options are very expensive since they are already at the limit of being Deep-In-The-Money. Therefore, a Call and a Put, respectively Delta 25 and -25, are “au pied du Gamma ». This means that the option buyer will be able to take advantage of the acceleration of the Gamma (especially between Delta 28 and 32) and thus be able to claim to sell the option much more expensive when it approaches the ATMF price because he will be selling a much higher Gamma.
Finally, we can see that Gamma evolves differently depending on the volatility of the underlying asset. High volatility will cause a flattening of the Gamma distribution curve according to the Strike Price. At low volatility, on the contrary, the distribution of Gamma will be more pronounced between a Strike Price at ATMF price and at the wings. Recalling that Gamma is the derivative of Delta, we can explain the fact that Delta is more sensitive for an underlying with low volatility. The sensitivity of the Gamma by volatility is also obtained by deriving the function, it can be represented by the Greek letter « Zomma ». The Gamma is therefore a Greek letter that is very popular among Market Optionners. Delta neutral management will have a different payoff depending on the Gamma of our portfolio. We must therefore be very careful to manage our Gamma so as not to be too sensitive to market variations while optimizing the PnL (profit and loss) of the portfolio.
Written on 08/12/2020
