Hierarchy
Vega → Vomma → Ultima.
Black–Scholes & Greeks · BSM-21 · Advanced
Ultima
What does Ultima measure, in which units, and what does it not predict?
Question it answers
How stable is my volatility convexity during another vol move?
Use
Mostly relevant in portfolios with nonlinear volatility exposure.
Perspective
Usually less important than Delta, Gamma, Theta, and Vega for basic trading.
What you will learn and prerequisites
- Explain Ultima in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
Ultima is the next volatility derivative after Vomma. It describes how volatility curvature changes again. Its purpose is understanding nonlinear exposure, not requiring a third-order calculation for every simple directional trade.
The sequence is Vega, changing Vega, then changing volatility curvature.
Read the derivative as a sentence
Three point-scaling factors produce the displayed per-point-cubed convention. In a price expansion the third-order term is one-sixth Ultima_raw×dSigma³.
Nested finite differences can amplify cancellation and bump-size errors. An analytic formula or several converging bump choices is better evidence than one arbitrary nested difference.
Use it within its scope
Large shocks need full repricing and a defensible volatility surface. Skew, term structure and exercise features may dominate model error before adding another Taylor term helps. High derivative order is not automatically high decision quality.
The model boundary
The smooth formulas require positive spot, strike, volatility and remaining time. Zero time and zero volatility still admit explicit value limits, but smooth Greek availability needs separate rules. No invalid or zero input is silently replaced with a default. Continuous dividends, constant parameters and European exercise are modeling assumptions, not claims about every real option.
One complete numerical experiment
The shared fictional European call uses spot 100, strike 100, 30 calendar days, sigma 25%, continuous rate 4% and yield 1%. Its BSM value is 2.977574155; d1=0.070239317 and d2=-0.001433455. Values are per option unit before contract multiplication.
Raw Ultima is -0.881871678693. The displayed convention gives -0.000000881872 Vomma-per-point² change per volatility point. Isolate IV +1 volatility point while holding every other input fixed.
Starting Vomma per volatility-point squared is -0.000000459114. The local estimate after the shock is -0.000001340986; a full-model recomputation gives -0.000001292425. Approximation minus recomputation is -0.000000048561. The difference shows the limitation of a local derivative, not a claim that the market must follow either number.
The graph varies only its labeled horizontal input. It is a series of controlled states, not historical observations or a future trajectory. Read its units before interpreting sign or magnitude.
Open the calculation without leaving this lesson
Ultima across controlled states
Vertical: Vomma-per-point² change per volatility point
Horizontal: IV (%)
1. Fictional example
Read the graph as a data table
| IV (%) | Fictional example |
|---|---|
| 10 | -2.54081e-05 |
| 12 | -1.23974e-05 |
| 14 | -6.80712e-06 |
| 16 | -4.09e-06 |
| 18 | -2.64258e-06 |
| 20 | -1.81489e-06 |
| 22 | -1.31407e-06 |
| 24 | -9.96746e-07 |
| 26 | -7.87819e-07 |
| 28 | -6.45731e-07 |
| 30 | -5.46375e-07 |
| 32 | -4.75206e-07 |
| 34 | -4.23138e-07 |
| 36 | -3.84328e-07 |
| 38 | -3.54912e-07 |
| 40 | -3.32281e-07 |
| 42 | -3.1463e-07 |
| 44 | -3.00692e-07 |
| 46 | -2.89559e-07 |
| 48 | -2.80572e-07 |
| 50 | -2.73245e-07 |
| 52 | -2.67216e-07 |
| 54 | -2.62211e-07 |
| 56 | -2.58021e-07 |
| 58 | -2.54485e-07 |
| 60 | -2.51478e-07 |
A correct number can be misread
The stated unit is Vomma-per-point² change per volatility point. Mixing raw decimal derivatives with point shocks or calendar time with remaining time can magnify or reverse the result.
The controlled comparison also omits live spread, surface dynamics, unsupported exercise effects and execution. A local sensitivity is not a market forecast or a guarantee of realized P/L.
Check your understanding
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
What is the displayed Ultima in this example?
Reveal the worked answer
-0.000000881872 Vomma-per-point² change per volatility point.
Always state the unit.
Does model agreement prove the same live-market change?
Reveal the worked answer
No.
Actual inputs, instrument features and executable quotes can differ.
Translate Ultima into its specific input, affected quantity and unit before applying it. Revalue when the local approximation no longer fits.
Try it in the working lab
- Inspect Vomma and Ultima for the same fictional state.
- Change volatility slightly, then substantially.
- Explain why a local third derivative is not a complete shock model.
Your live inputs are independent of the fixed worked example above. Return to the “Lesson example” preset to reproduce it.
My reading record
Personal review marks are not quiz passes, course certification or trading approval. Old plugin progress is not overwritten.
Sources, assumptions and downloadable script
External references support definitions and conventions. Worked scenarios and prose are original. No current quotes or dealer inventory are claimed.
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