The Gamma of an option is the partial derivative of the options Delta with respect to the price of the underlying security. So the Gamma measures the sensitivity of the option Delta to the price of the underlying security. The Gamma of a call option is therefore given to us by Delta_2C, Delta S squared. Again, it's somewhat tedious, but it can easily be calculated using basic calculus. We can take the partial derivatives of the Black-Scholes formula to calculate the Gamma. If we do that, we will find that is equal to this expression here, e to the minus c times T, N of d1 divided by Sigma S square root T. How about the Gamma of a European put option? Well, that's easily calculated from put-call parity. So put-call parity is given to us here. So therefore, we can actually say that P is equal to C plus e to the minus rT times K, minus e to the minus cT times S. So we can therefore see that Delta_2P, Delta S squared is equal to Delta_2C Delta S squared. Well, plus 0 minus 0 because the second partial derivative of this is equal to 0, and the second partial derivative of this with respect to S is equal to 0. So therefore, we see that the Gamma for a put option is equal to the Gamma for a call option. So once we know the Gamma for European call option, we therefore have the Gamma for European put option. In fact, you can see that this expression is always greater than or equal to zero. So the Gamma for European options is always positive. This is due to what's called option convexity. Here is the plot of the Gamma for European options is as time to maturity varies. So the Gamma here is a function of the stock price, and you've got three different times to maturity, 0.05 years, 0.25, and 0.5 years as we saw before. Notice that the Gamma is steepest for the shortest maturity. So in this case, for T equals 0.05 years, i.e. approximately 2.5 weeks to maturity, we see that the Gamma is very steep around the strike of K equals 100, but we also see that it falls away to zero much faster than the option when T equals 0.25, or the option when T equals 0.5 years. The reason is as follows, that Delta of the European option when T equals 0.05 years, well, it's going to be a half or approximately a half when the stock prices at the strike K, but as the stock price goes up, the Delta's going to move towards one. It's going to move towards one much faster than the options with the higher time to maturity. Similarly, as the stock price falls below the strike of 100, the Delta of the call option is going to move towards zero, and it's going to move towards zero much faster than the Delta of the options with times to maturity of 0.2 and 0.5 years. So this plot here is just another way of looking at this plot. In the option when T equals two-and-a-half weeks or 0.05 to maturity, we see that when S equals 100, the Delta is approximately 0.5. But for small moves of S above 100, the Delta quickly goes towards one , and for small moves of S below 100, the Delta quickly goes towards zero, and it does so much faster than the options with larger times to maturity. So we're seeing that the option, which is two-and-a-half weeks to maturity has a much higher Gamma then the option when T equals 0.5 years or T equals 0.25 years to maturity. Another way of looking at this is and now plot the Gamma as a function of time to maturity. So one year to maturity, six months to maturity, zero time to maturity. We see that for the option that's out of the money, 10 percent out of the money, or 20 percent out of the money, this would also be true for options that are in the money, the Gamma of those options actually goes towards zero. It falls towards zero. That's because as the time to maturity goes to zero, we know for sure we're not going to be exercising if we're out of the money, and we know for sure that we are going to be exercising if we're in the money. In other words, Delta will be one if we're in the money, Delta will be zero for out of the money, and so Gamma will be zero in both cases. On the other hand, if we're dealing with an up the money option, where the current stock price is equal to K. So here is where S is equal to K for this blue curve. Well, then as the time to maturity goes to zero, we're going to find that our Delta is equal to a half, but that the Gamma will actually be very, very large and grow very large, and that's because small moves, and as we move the delta to either one if S increases, or zero if S decreases. So we get a very large Gamma for at-the-money options. I should mention as well by the way that with all have these plots that we're looking at a Delta and Gamma, market practitioners understand this behavior, they understand it at an intuitive level. They know how the Delta of a call on European put option behaves and how the Gamma of these options behave. So it's very important that if you're working with options in practice, you understand these figures, and that you understand why they behave and look the way that they do.
