Vomma

Definition

Vomma is a second-order “Greek” that measures how an option’s vega changes as the implied volatility (σ) of the underlying changes. Formally:
Vomma = ∂ν / ∂σ = ∂²V / ∂σ²
where V is the option value and ν (vega) is the sensitivity of the option price to volatility.

How vomma works

• Vomma captures the convexity of vega with respect to volatility. A positive vomma means vega increases when volatility rises and decreases when volatility falls; a negative vomma implies the opposite.
• Vomma provides insight beyond vega alone: vega tells how price reacts to small volatility moves, while vomma indicates how that sensitivity itself will shift if volatility changes further.
• Vomma tends to be larger for options with higher vega — typically near-the-money and longer-dated options — but its sign and magnitude depend on moneyness, time to expiry, and volatility.

Key formula(s)

• Vega (Black–Scholes): ν = S · φ(d1) · √t
  with φ(d1) = (1/√(2π)) e^(−d1²/2) and
  d1 = [ln(S/K) + (r + σ²/2) t] / (σ √t)

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• Vomma (convenient form using vega):
  Vomma = Vega · (d1 · d2) / σ
  where d2 = d1 − σ √t

These expressions use:
– S = underlying price
– K = option strike price
– r = risk-free interest rate
– σ = volatility (implied)
– t = time to expiry (in years)
– φ = standard normal probability density function

Practical interpretation for traders

• Long options: generally prefer positive vomma because rising volatility increases both option value and vega, amplifying gains from further volatility increases.
• Short options: generally prefer negative vomma (or low positive vomma) because rising volatility would otherwise increase vega and increase the risk of larger losses.
• Risk management: traders who hedge vega (vega-neutral) should also monitor vomma because changes in volatility will alter the vega exposure; rebalancing may be required after large volatility moves.

Example: if an option has vega = 5 and vomma = 0.2, a 1-percentage-point rise in implied volatility will increase vega by about 0.2 (vega → 5.2), changing subsequent sensitivity to further volatility moves.

Relationship to pricing models

Vomma is a higher-order input when assessing option price behavior under models like Black–Scholes. Including vomma helps gauge how robust hedges and positions will be under nonstationary volatility environments.

Key takeaways

• Vomma measures how vega changes with volatility — it is vega’s convexity.
• Positive vomma magnifies the effect of rising volatility on option sensitivity; negative vomma does the opposite.
• Traders use vomma alongside vega and other Greeks to manage volatility risk and hedge dynamically.