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Ultima Greek: How Vomma Changes with Implied Volatility

Understand Ultima as a local third-order model sensitivity to a volatility input, normalize its units, and verify it with finite differences.

Updated

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Ultima measures how an option’s Vomma, also called Volga, changes when a chosen volatility input changes, with spot, time, rates, dividends, and all other model inputs held fixed. Under the convention used here, if model 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 can show how a second-order volatility approximation bends as the selected input changes, but it does not predict IV, guarantee a profit, or describe every change in a live volatility surface. A surface-based result also depends on which nodes move and what is held fixed.

Scope: this page explains a theoretical sensitivity, not the terms or price of any listed or OTC product and not an account-level risk calculation. The example is a European-style Black-Scholes model output, not an executable quote. The OIC and OCC materials concern U.S. exchange-traded options; product terms, broker rules, taxes, and law can differ by market, account, jurisdiction, and date. Check current official documents and qualified advisers. This page does not provide individualized investment, legal, or tax advice.

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.

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 the Black-Scholes Vomma formula, Vomma is approximately 26.2235 at 29% IV and 21.6499 at 31% IV, quoted per unit of decimal volatility squared. A centered finite-difference estimate at 30% is:

Ultima ≈ (21.6499 − 26.2235) ÷ (0.31 − 0.29) = −228.7

That is about −228.7 dollars per share per unit of decimal volatility cubed under this convention, or approximately −0.0002287 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.

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.

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.

Authoritative sources

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