Vanna

The Vanna is the sensitivity of the option vega to the underlying spot level. In mathematical terms it is the second derivative of the option price with respect to the implied volatility and the underlying spot. In simple terms it tells by how much the option vega changes when the underlying spot changes or equivalently it tells by how much the option delta changes when the implied volatility changes.

Here is a chart of the Vanna:



It shows that for out of the money calls the Vanna is positive and for in the money calls the Vanna is negative with various convexity effects.

Intuitively, the Vanna is a measure of how much the Delta will change when the implied volatility changes. Let’s look at a numerical example:

						Spot		1300
						Strike		1400
	Volatility	Delta		Vanna		Maturity		0.5
	15.0%	0.2720117				Volatility		15.0%
	16.0%	0.2868161				Interest Rates		3.0%
						Dividend yield		2.0%
	Difference	0.0148044

For this out of the money call, the delta of the option goes up from 27.2% to 28.6%. In practice this is important because if the delta changes then the option trader needs to re-hedge his delta in the market.

						Spot		1500
						Strike		1400
	Volatility	Delta		Vanna		Maturity		0.5
	15.0%	0.7658696				Volatility		15.0%
	16.0%	0.7537522				Interest Rates		3.0%
						Dividend yield		2.0%
	Difference	-0.012117

Alternatively, for an In the money call, the delta of the call option decreases from 76.5% to 75.3% when the implied volatility goes up from 15% to 16%.

Vanna in Practice

For the individual investor the Volga is one of the least important Greeks as for the individual investor his position will be mostly dominated by the delta.

The importance of the Vanna for the professional option trader is due to the fact that his delta hedging will be influenced by the Vanna. For instance, in the example above for the Out of the Money call option, the delta goes up from 27.2% to 28.6% when the volatility goes up from 15% to 16%. This means that if the position was initially delta neutral it will now show an excess of delta, and the delta neutral trader will need to sell the underlying asset in the market to be flat delta.

Market makers are always aware of their Vanna because they know that any change in the implied volatility will make them buy or sell in the market futures contracts or cash.

Vanna: sensitivity of vega to changes in the underlying