The Delta of an option measures the sensitivity of the option to price changes in the underlying asset, . The Vanna of an option measures Delta's sensitivity to volatility, sigma. The following example illustrates the characteristics of the Vanna of an option with respect to these two variables.
In this example, the Vanna is defined as a function of the underlying asset price , and volatility, sigma. For a European call option, we will assume that the strike price is 100, time to maturity is 1, 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 Vanna behaves as a function of the risk-free interest rate, the dividend yield, and volatility. To compute the Vanna 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: