Third-order Greek · ∂³V/∂σ³

Ultima: the third derivative of an option to implied volatility

Ultima, sometimes written DvommaDvol, is the change in vomma for a change in implied volatility. It's the outermost of the commonly named Greeks, and almost nobody needs it day to day.

The formula

d₁ = [ ln(S/K) + (r + σ²/2)·T ] / (σ√T)
d₂ = d₁ − σ√T

Ultima = ∂Vomma/∂σ = −(Vega / σ²)·[ d₁·d₂·(1 − d₁·d₂) + d₁² + d₂² ]

Black-Scholes, European options, no dividends. S spot, K strike, T years to expiry, r the risk-free rate, σ implied volatility, N(·) the standard normal CDF and φ(·) its density.

Where it shows up

Ultima tells a volatility trader how their vomma will change after a large move in implied vol, which matters for books of far out-of-the-money options during a volatility shock. It's small and noisy everywhere else.

The family, in order

Vega (∂V/∂σ) → vomma (∂²V/∂σ²) → ultima (∂³V/∂σ³). The same ladder for price is delta → gamma → speed.

SPX ATM implied volatility, skew through the session and the skew smile in Zanshin Flow — SPX implied volatility and skew through the session.

Questions

What is ultima in options?

Ultima is the third derivative of an option's value with respect to implied volatility, equivalently the rate of change of vomma with volatility.

Do traders use ultima?

Rarely. It matters mainly to volatility traders holding large books of far out-of-the-money options through big volatility moves.

How does ultima relate to vega?

Vega is the first derivative of value to implied volatility, vomma the second and ultima the third.

Not financial advice. This article explains how options and dealer hedging work. It is not a recommendation to buy or sell anything.