Analysing the role of individual Greeks in derivatives and structured products is important. In this article, we continue our exploration of different Greeks and examine “Vanna”.

Vanna is a second-order Greek, defined as the rate of change of Delta with respect to volatility, or equivalently, the rate of change of Vega with respect to the underlying spot.

It is a useful Greek for traders as part of their risk management toolkit as it gives insight into the relationship between spot and volatility on option prices. These two quantities are both critical to an option's value, and are closely related since the volatility level determines the likely scale of future spot moves.

The exact definition of Vanna, denoted V(S) is given by:

Where  is volatility, S is the spot price and f(..) is the option price function. This second-order derivative measures the change in price with respect to both volatility and spot. As with all mathematical partial derivatives, the order of evaluation can be reversed:

This explains the two alternative definitions mentioned above, the rate of change of Vega with respect to spot and the rate of change of Delta with respect to volatility.

Vanna exhibits interesting behaviour, for which we can examine some examples. For a call option with two-year maturity, the chart below shows the values of Delta, Gamma and Vanna, rescaled as necessary:

Delta rises steadily from zero to one, and Gamma peaks near at-the-money level, representing the greatest rate of change of Delta. Both of these effects are well known since Delta has the highest gradient near at the money, where the option changes rapidly from being in the money or out of the money (which is strongly tied to the level of delta).

Understanding positive and negative Vanna

As shown in the chart, Vanna takes both positive and negative values and is the only Greek of these three to do so. For low spot values, when the option is out-of-the-money, an increase in volatility raises the option’s price and a higher chance of finishing in the money. Since the option has zero intrinsic value, the price increase must come from a higher chance of a positive outcome.

The situation for higher spot values is less intuitive. A rise in volatility still increases the option price (a call option always has positive Vega), but the price gain comes from a higher potential payoff due to larger upward moves. However, Delta may decrease in this case because the increased volatility also raises the probability of a significant downward move, which potentially taking the option out of the money. This reduction in Delta is what gives the negative Vanna.

For shorter-dated options, Vanna is lower, as shown in a second chart.

The positive and negative zones persist, but while Gamma has increased, Vanna has gone down. This aligns with the interpretation of Vanna as the rate of change of Vega with respect to spot. Since Vega is much lower for shorter maturities, and accordingly Vanna drops too. Vega measures the effect of “future” volatility, whereas Gamma tracks changes caused by current moves.

Vanna is important because spot and volatility are arguably the two pricing parameters that require the most dynamic hedging. Generally, volatility increases when spot decreases and can often be observed when VIX values jump after the market falls.

A call option exhibits positive Vanna for lower spot values. In the scenario of market falls and volatility rises (or market up and volatility down), this second-order change can cause a loss for a hedged Delta-neutral and Vega-neutral position. In structured product markets, issuers commonly sell embedded call options and therefore, they would make money in such scenarios. However, they are more exposed when the underlying is above the initial levels and the valuation is higher.

Model dependence and calibration sensitivity

Market participants use a variety of complex option models, such as stochastic volatility and local volatility models. Depending on how these are calibrated product prices will vary somewhat and this translates to different views of Vega. Consequently, Vanna exposure is partly model-dependent and potentially sensitive to different choices made.

This spot-volatility relationship measured by Vanna becomes particularly relevant during periods of market stress, when volatility spikes can dramatically alter the risk profile of options portfolios. Traders use Vanna to anticipate how their Delta hedges will need to be adjusted in response to changing volatility conditions.

Second-order Greeks are the most complicated to calculate and interpret, but they are critical as they indicate rebalances that are needed. Just as Gamma gives rise to changes in Delta hedges due to spot values, Vanna simultaneously shows hedging changes in Delta and Vega.

Structured products often combine multiple options with different strikes, expirations, and underlying assets, creating complex Vanna exposures that require careful management. These products may include features like autocalls, barriers and exotic payoffs that create non-linear sensitivities to market movements.

In structured products linked to multiple underlying assets, Vanna becomes even more complex due to correlation effects and cross-sensitivities. While diagonal terms are most important (changes due to spot and volatility of the same underlying), it may also be useful to measure cross-Vanna terms (spot of one asset and volatility of another) may also be relevant.

Many structured products include barrier features or path-dependent payoffs. These will impact Vanna, just as they affect other Greeks. The second-order nature of Vanna means it responds rapidly to market changes and is more difficult to calculate numerically.

Hedging Vanna exposures presents unique challenges. Traders typically use combinations of options with different strikes and expirations, creating complex hedge strategies that require constant monitoring and adjustment.

Vanna represents a critical component of sophisticated derivatives risk management. It provides insights into product and portfolio behaviour that first-order Greeks or Gamma alone.