The Greeks and Option Risk Management › Higher-Order Greeks

Beyond the standard five Greeks, several higher-order sensitivities become important for sophisticated risk management. These cross-partial derivatives capture how the primary Greeks change when other market variables move, and gives a more complete picture of option risk dynamics.

Vanna measures how delta changes when volatility changes:

$$\text{Vanna} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial \nu}{\partial S}$$

where:

• Vanna: sensitivity of delta to volatility
• $\Delta$: delta
• $\sigma$: volatility
• $\nu$: vega
• $S$: spot price

This cross-partial derivative matters when both spot and volatility move together, as they often do in equity markets (where volatility tends to spike when prices fall). The equality between $\frac{\partial \Delta}{\partial \sigma}$ and $\frac{\partial \nu}{\partial S}$ follows from the symmetry of mixed partial derivatives, a consequence of the smoothness of the Black-Scholes pricing function.

For European options under Black-Scholes:

$$\text{Vanna} = -\frac{d_2}{\sigma} N'(d_1) = \frac{\nu}{S}\left(1 - \frac{d_1}{\sigma\sqrt{T}}\right)$$

where:

• $\text{Vanna}$: sensitivity of delta to volatility
• $d_1, d_2$: standardized moneyness terms
• $N'(d_1)$: standard normal PDF
• $\nu$: vega
• $S$: spot price
• $\sigma$: volatility
• $T$: time to expiration

Volga or vomma measures the convexity of option value with respect to volatility:

$$\text{Volga} = \frac{\partial^2 V}{\partial \sigma^2} = \frac{\partial \nu}{\partial \sigma}$$

where:

• Volga: sensitivity of vega to volatility
• $V$: option value
• $\nu$: vega
• $\sigma$: volatility

This becomes important when volatility itself is volatile. A portfolio with high volga benefits when realized volatility of implied volatility is high. In markets where implied volatility swings dramatically, volga exposure can dominate P&L.

Charm measures how delta changes as time passes:

$$\text{Charm} = \frac{\partial \Delta}{\partial t} = \frac{\partial \Theta}{\partial S}$$

where:

• Charm: sensitivity of delta to time
• $\Delta$: delta
• $t$: time
• $\Theta$: theta
• $S$: spot price

This is important for understanding how your hedge will need to change even if the spot price remains constant. A position that is delta-neutral today may drift away from neutrality simply due to the passage of time, requiring rebalancing.

Speed is the third derivative of option price with respect to spot:

$$\text{Speed} = \frac{\partial \Gamma}{\partial S} = \frac{\partial^3 V}{\partial S^3}$$

where:

• Speed: sensitivity of gamma to price
• $V$: option value
• $\Gamma$: gamma
• $S$: spot price

Speed matters when gamma itself is highly sensitive to price movements. For large portfolios of options with different strikes, understanding speed helps us anticipate how our gamma exposure will change during market moves.

We can calculate these higher-order Greeks using the formulas derived from the Black-Scholes partial derivatives. These values help quantify the stability of the primary Greeks: