How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega

Chapter 8Gamma

Gamma is the change of the delta of an option in relation to the change in the underlying Future.

Gamma ( for the capital letter) is the first derivative of delta, or the second derivative of the value of the option (as delta is the first derivative of the value of an option), mathematically: or , which actually doesn't say more than – it's the change of the delta relative to the change of the Future (or to make it more accurate: the change of the change of the value of the option relative to the change of the Future). Throughout this book the Greek letter γ or simply gamma is used for denoting the gamma. The formula for calculating it is as follows: where is the probability density function (for d1 see chapter on delta).

As shown in the chapter on delta, and depicted in Chart 8.1, the change in the delta of a 50 call option could bring a large profit by each time hedging the delta position. When the market moves up (e.g. from 50 to 54), the owner of the 50 calls can ...