The Vega of an option measures the sensitivity of the option to volatility, sigma. The Vomma of an option measures Vega's sensitivity to volatility. The following example illustrates the characteristics of the Vomma of an option with respect to volatility as well as the time to maturity.
In this example, the Vomma is defined as a function of volatility, sigma, and time to maturity, T. For a European call option, we will assume that the strike price is 100 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 Vomma behaves as a function of the risk-free interest rate, the dividend yield, and volatility. To compute the Vomma 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: