The Gamma of an option measures the sensitivity of the Delta to changes in the value of the underlying asset, . The Speed of an option measures Gamma's sensitivity to changes in the value of the underlying asset. The following example illustrates the characteristics of the Speed of an option with respect to changes in the value of the underlying asset as well as movement in the time to maturity.
In this example, the Speed is defined as a function of the underlying asset price , and time to maturity, T. For a European call option, we will assume that the strike price is 100, volatility is 0.10, and the risk-free interest rate of 0.05. We also assume that this option does not pay any dividends.
We can also see how the Speed behaves as a function of the risk-free interest rate, the dividend yield, and volatility. To compute the Speed of a European call option with strike price 100 maturing in 1 year, we take:
This can be numerically solved for specific values of the risk-free rate, the dividend yield, and the volatility.
It is also possible to use the generic method in which the option is defined through its payoff function:
Here are similar examples for the European put option: