# Ultima Greek: How Vomma Changes with Implied Volatility

Understand Ultima as the third derivative of option value with respect to implied volatility, normalize its units, and verify it with finite differences.

Canonical: https://wiki.fcontext.com/options/ultima-greek/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

**Ultima** measures how an option's Vomma changes when implied volatility changes, with spot, time, rates, dividends, and the model otherwise held fixed. If option value is `V` and decimal volatility is `σ`, then:

`Vega = ∂V/∂σ`, `Vomma = ∂²V/∂σ²`, and `Ultima = ∂³V/∂σ³ = ∂Vomma/∂σ`.

Ultima is a local third-order model sensitivity. It helps diagnose whether a second-order volatility approximation may bend further during a large IV shock. It does not predict IV, guarantee a profit, or describe every change in a live volatility surface.

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## Mechanism and units

A Taylor approximation around the current volatility is:

`ΔV ≈ Vega×Δσ + ½×Vomma×(Δσ)² + ⅙×Ultima×(Δσ)³`

The expansion isolates volatility effects only. Delta, Gamma, Theta, rates, dividends, skew, and execution still matter. Because Vega and Vomma change after every shock, full repricing is preferable when `Δσ` is large.

Units are a major source of error. A model using decimal volatility treats `30%` as `0.30`; a screen may report sensitivity per one volatility point, where `30%→31%` is one point. Converting a decimal-volatility Ultima to a per-point-cubed figure divides it by `100³ = 1,000,000`. Some vendors instead report the change in their displayed Vomma for a one-point IV move. Always identify the convention before comparing values.

Ultima can be positive or negative and can change sign with moneyness, maturity, rates, dividends, and model choice. Its sign says how local Vomma changes as the chosen IV input rises; it is not a bullish or bearish signal.

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## Finite-difference example

Consider a European Call with stock price `$110`, strike `$100`, six months remaining, a continuously compounded risk-free rate of `4%`, no dividend, and Black-Scholes assumptions. Reprice it at IVs of `29%`, `30%`, and `31%`.

Using small volatility bumps to estimate the second derivative, suppose Vomma is approximately `26.22` at `29%` IV and `21.65` at `31%` IV, quoted per unit of decimal volatility squared. A centered estimate at `30%` is:

`Ultima ≈ (21.65 − 26.22) ÷ (0.31 − 0.29) = −228.5`

That is about `−228.5` dollars per share per unit of decimal volatility cubed under this convention, or approximately `−0.0002285` dollars per share per volatility-point cubed after dividing by one million. The negative result means local Vomma falls as IV rises around this particular input set.

The calculation is a model-unit check, not a trade forecast. Different bump sizes, American exercise, dividends, an IV surface, rounding, or a different model can produce another value. For one standard 100-share contract, apply the multiplier only after confirming that the quoted Greek is per share.

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## Risks and practical controls

- Record spot, strike, expiry timestamp, exercise style, IV surface, rates, dividends, borrow, model, bump size, and units with every observation.
- Verify a displayed Ultima by bumping IV up and down, recalculating Vomma, and dividing by the total decimal-volatility interval.
- Test several bump sizes. A result that changes dramatically may be numerical noise or evidence that the local approximation is unstable.
- Aggregate signed exposure as `per-share Greek × signed contracts × multiplier` only after normalizing every leg to the same convention.
- Shock individual strikes and expirations rather than assuming the whole volatility surface moves in parallel.
- Reprice the entire position for large moves, event risk, short expiries, deep moneyness, or positions with early-exercise features.
- Include Bid/Ask spreads, liquidity, assignment, margin, and discrete hedge sizes; a mathematically precise Greek does not make a hedge executable.
- Treat model disagreement as uncertainty. High-order derivatives amplify small errors in prices, curves, dividends, and surface interpolation.

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## Common misconceptions

- **“Ultima is the rate at which IV changes.”** It is the sensitivity of Vomma to an assumed IV change, not an IV forecast.
- **“A negative Ultima means the option should be sold.”** The sign describes local curvature under a model, not expected return.
- **“Ultima replaces Vega and Vomma.”** It is an additional term; first- and second-order exposures usually dominate small moves.
- **“Values from two platforms are directly comparable.”** Decimal versus point scaling, per-share versus per-contract units, models, and bump methods can differ.
- **“One Ultima summarizes a portfolio.”** Each strike and expiry can face a different surface shock; gross leg risk can remain large when net exposure is small.
- **“Adding more Greeks removes model risk.”** A higher derivative can be less stable and more sensitive to input and numerical error.

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## Related topics

- [Charm, Vanna, and Vomma](/options/charm-vanna-vomma/)
- [Vega](/options/vega/)
- [Implied Volatility](/options/implied-volatility/)
- [Option Model Risk](/options/option-model-risk/)

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## Authoritative sources

- [The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062) — Fischer Black and Myron Scholes
- [Theory of Rational Option Pricing](https://doi.org/10.2307/1831029) — Robert C. Merton
- [Vega](https://www.optionseducation.org/advancedconcepts/vega) — Options Industry Council
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — Options Clearing Corporation