Vega and Vomma

Of all the things that move an option's price, volatility is the one option traders obsess over. Vega measures the option's sensitivity to it: how many dollars the option gains or loses when volatility changes by one point. Vomma (also called volga) goes one level deeper, it measures how vega itself changes as volatility moves. If vega is your speed, vomma is your acceleration.

Why bother with the second one? Because vega isn't a constant. An option that's very sensitive to vol today can become much more sensitive after vol rises. If you only track vega, a big move in volatility will surprise you, your exposure grew while you weren't looking. Vomma is the Greek that tells you how much it will grow.

Vega: the first-order sensitivity

Vega is the change in the option's value $V$ for a change in volatility $\sigma$:

$$\text{vega} = \frac{\partial V}{\partial \sigma} = S\, n(d_1)\sqrt{T}.$$

Read the pieces: $S$ is the spot price, $n(d_1)$ is the standard normal bell-curve height at $d_1$ (largest when the option is at the money), and $\sqrt{T}$ is the square root of time to expiry. Two facts fall straight out. Vega is largest for at-the-money options, and it grows with the square root of time, so long-dated options carry far more vega than short-dated ones. Every long option, call or put, has positive vega: more volatility means more chance of a big favourable move, which is worth more.

Vomma: the curve in the sensitivity

Vomma is the change in vega as volatility moves, the second derivative of value with respect to vol:

$$\text{vomma} = \frac{\partial^2 V}{\partial \sigma^2} = \text{vega} \cdot \frac{d_1 d_2}{\sigma}.$$

The useful part is the sign of $d_1 d_2$. For an at-the-money option, $d_1$ and $d_2$ sit on opposite sides of zero, so their product is negative or tiny, vomma is near zero, and vega barely changes as vol moves. Value is almost a straight line in volatility. For an out-of-the-money option (the "wings"), $d_1$ and $d_2$ share a sign, so $d_1 d_2 > 0$ and vomma is positive: as vol rises, vega rises too, and value curves upward. Those wing options are long vomma, they are a bet on volatility itself becoming volatile.

Worked example

Take a one-year at-the-money call, spot $S = 100$, volatility $\sigma = 20\%$. Here $n(d_1) \approx 0.397$, so

$$\text{vega} = 100 \times 0.397 \times \sqrt{1} = 39.7.$$

That's per one full unit of vol, so per one percentage point (0.01) it's about $39.7 \times 0.01 \approx 0.40$. If implied vol jumps from 20% to 22%, this option gains roughly $2 \times 0.40 = 0.80$ in value. Because it's at the money, its vomma is small, so that 0.40-per-point sensitivity stays put whether vol is 20% or 22%.

Now compare a deep out-of-the-money call on the same stock. Its vega might be a smaller 0.15 per point at 20% vol, but it's long vomma. Push vol to 30% and that vega might climb to 0.28, nearly doubling. The wing option's exposure to volatility feeds on itself, which is precisely why traders reach for out-of-the-money options when they expect a genuine volatility explosion, not just a drift higher.

Where it matters and where it misleads

Vomma is the reason the volatility smile has curvature. Out-of-the-money options are long vomma, and traders demand a higher implied vol for them precisely to be compensated for that convexity, bending the smile upward at the wings. Products built to be pure vol bets, like variance swaps, are essentially engineered to hold constant vega across strikes, which requires stacking up exactly this wing exposure.

• Vega is not additive across dates. A book neutral in total vega can still be badly exposed if front-month and back-month vols move differently (the term structure twists). Bucket vega by expiry.
• Vega assumes vol actually moved the way you modelled. In a real smile, when spot moves, the implied vol you'd use moves too (the vanna effect), so a "pure vega" number quietly mixes in spot risk.
• Ignoring vomma understates tail vol risk. A short-wing position looks cheap on vega alone, then vol spikes, vega balloons against you, and the loss is far worse than the vega number suggested.

Vega and vomma sit inside the full family of The Option Greeks, trade off against theta, and are what a gamma scalper is implicitly long when they own options.