Ultima Calculator
The formula
What the ultima tells you
The third derivative with respect to volatility — the “vol of vol” exposure hiding inside plain options. For most strategies it is academic; for anyone holding convexity through a regime change it quietly owns part of the P&L.
Calculation: Ultima = ∂Volga/∂σ — third vol derivative (∂³V/∂σ³). Enter your inputs and every substitution is shown below the result.
Reading the result
If ultima is large, your volga estimate is a moving target: vega P&L will accelerate (or decelerate) nonlinearly as the stress unfolds. It explains part of why vol traders’ realized P&L misses their vega math.
Rules of thumb: Meaningful only when σ can move 10–20+ points: event vol, crisis regimes, vol-of-vol products
Limitations and best practice
With 60-day options the entire vol-Greek stack compresses; with 5-day options surface noise dominates estimation. Ultima is the most model-dependent number on this page — its honesty depends entirely on your vol surface.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
Does ultima appear in risk systems?
Is ultima how VVIX relates to options?
Can I ignore it?
Why do long straddles out-earn vega in crises?
Source & methodology
Jaubert et al., Advanced Greeks (2011).
How Ultima works — and what your result really means
You are here because you need a specific number, and you want to see how it was produced before you rely on it. Searchers land on an Ultima page for three reasons — to plan, to compare options, and to check that somebody else’s arithmetic is honest. This tool serves all three, and everything after the calculator explains the model instead of hiding it.
Most confusion about Ultima comes from mixing up the input you control with the input you merely observe. The formula treats them identically, but your decisions should not: one deserves measurement, the other deserves a defensible estimate.
What makes Ultima worth calculating is not precision but position: it tells you which side of a threshold you are on, and by how much. A rough figure that settles the direction beats an elaborate one that never commits.
Every figure in the prose is generated by the engine rather than written beside it — the worked example resolves to ultima per 1%³ vol (×100,000) = 8.880 on the default inputs, and the sensitivity section ranks the inputs that move the result — Strike first, at ≈207.4 % of the headline for a ten-percent error either way in that input. Work down the page and by the end you could re-derive the result in a spreadsheet from memory.
Inside the Ultima calculation: formula and every input
Here is the complete model this tool runs — Ultima = ∂Volga/∂σ — third vol derivative (∂³V/∂σ³) — with each input explained below. If a competitor’s page gives you a different answer from the same inputs, this is the text that settles which of you is right.
Underlying price ($) (example value 210) is a quantity rather than a category — precision here is cheap and worth having. Keep every money input in the same currency; mixing systems quietly corrupts the comparison.
Strike ($) (example value 108) is a quantity rather than a category — precision here is cheap and worth having. If you are unsure what belongs in this field, look at the printed steps: the substitution shows exactly how the engine uses it.
Time to expiry (years) (example value 1) is a quantity rather than a category — precision here is cheap and worth having. Use the value as of today, not as of the last time you looked — time fields drift silently.
Risk-free rate (%) (example value 5) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Enter it exactly as your document states it — quoted annual, monthly and effective rates are three different numbers.
Implied volatility (%) (example value 26) is a quantity rather than a category — precision here is cheap and worth having. Check the convention first: if the source says “6.5 % per year”, type 6.5 and let the formula do the rest.
Option type (example value Call) is a categorical input — the option you choose decides which numbers the formula uses. This input feeds the formula directly; a defensible figure here is worth more than false precision anywhere else.
Ultima on the example inputs: the complete substitution
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Underlying price ($), Strike ($), Time to expiry (years), Risk-free rate (%). The computed ultima per 1%³ vol (×100,000) comes out at 8.880, which the interpretation band labels “Volga flexes 8.880 per vol point (×100,000) — the curvature-of-curvature that only 10+ point vol regime shifts monetize”.
- Ultima = ∂Volga/∂σ per pt³ = 0.00008880 (×100,000 = 8.880)
- The deepest vol term: only relevant for straddle/vol-of-vol traders
- 10-pt regime move: volga ×10,000 2.068 → 10.947
Volga after +10 pts (×10,000) — 10.947. Read next to the headline, this is the figure that shows the shape of the trade-off.
Volga now (×10,000) — 2.068. This is the number experienced users quote when the headline alone would mislead.
Ultima ×100,000 per contract — 888.0. If your inputs change, this is usually the first companion figure to move.
Vol regime sensitivity rank — 4.29 ×. This is the number experienced users quote when the headline alone would mislead.
Overwrite any input above and every line of this example updates with it — the substitution is live, not a screenshot of somebody else’s numbers.
How far Ultima can reasonably move
Read this table as the practical error bar on the headline figure. It answers the question people actually have after seeing a single number: how wrong could this be if the main input is off by ten or twenty percent?
| Strike setting | Ultima per 1%³ vol (×100,000) |
|---|---|
| 80 % of the example value | 0.174 |
| 90 % of the example value | 1.706 |
| example inputs | 8.880 |
| 110 % of the example value | 27.300 |
| 120 % of the example value | 52.176 |
The corridor runs from 0.174 to 52.176 across that band — which is the range worth planning inside rather than the single 8.880 point estimate. Reproduce it in the form above by editing Strike alone; the substitution steps will follow your figures exactly.
Where Ultima is fragile — and where it is not
Nobody should have to guess which input matters. The engine was re-run once per field below — ten percent up, ten percent down, everything else frozen — and the results ranked by how far they moved the ultima per 1%³ vol (×100,000). Read it as a map of where Ultima is fragile — drawn for the example inputs, which is the map you want when deciding what to measure carefully.
- Nudge Strike by ±10 % and Ultima lands between 1.706 and 27.300 (sensitivity ≈207.4 %)
- Underlying price +10 % → Ultima = 2.233 · −10 % → 27.021 (the ultima per 1%³ vol (×100,000) shifts ≈204.3 %)
- Nudge Implied volatility by ±10 % and Ultima lands between 3.637 and 15.514 (sensitivity ≈74.7 %)
- Time to expiry +10 % → Ultima = 13.561 · −10 % → 5.203 (the ultima per 1%³ vol (×100,000) shifts ≈52.7 %)
- Nudge Risk-free rate by ±10 % and Ultima lands between 8.609 and 9.157 (sensitivity ≈3.1 %)
Strike is the dominant lever here: ten percent either way moves the result by about 207.4 %, from 1.706 to 27.300. Measure that input from a document rather than from memory, and Underlying price is effectively tied with it (≈204.3 % for the same nudge), so verify the pair together. The rest of the inputs earn only a rounding-error of attention, and that asymmetry is the practical payoff of measuring rather than asserting.
Interpreting the Ultima verdict
The value above only becomes useful the moment you know what it is being measured against. The verdict attached to the default calculation — “Volga flexes 8.880 per vol point (×100,000) — the curvature-of-curvature that only 10+ point vol regime shifts monetize” — is one band on a ladder your own figure may sit somewhere else on. How much of that band is luck? On Strike alone ten percent either way covers 1.706 to 27.300. The practical reading rules: ask what would have to change to move you one band. If the answer is “a small, plausible input difference”, the band itself is not the finding — the uncertainty is.
Where Ultima sits next to similar measures: this tool answers “how much is the Ultima”, while the related calculators answer “what do I do about it”. Reading the result alone tends to over-weight a single snapshot; pairing it with the companion tools above turns the number into a decision.
Common mistakes with Ultima (and the fix)
With 60-day options the entire vol-Greek stack compresses; with 5-day options surface noise dominates estimation. Ultima is the most model-dependent number on this page — its honesty depends entirely on your vol surface. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Using stale inputs. Ultima is only as current as the numbers behind it — an estimate carried over from a previous season, rate table or report quietly biases every output. Re-measure the inputs first; that single habit fixes more bad Ultima figures than any formula change.
One mistake is specific to this engine: assuming every input matters equally. The computation above shows Strike dominating — a careful figure there beats false precision everywhere else.
The subtlest error is answering the wrong question precisely. Confirm that ultima per 1%³ vol (×100,000) is actually the quantity your decision needs — the arithmetic is flawless either way, which is what makes the mistake invisible.
Six things to confirm before acting on Ultima
- Check that every field the formula needs actually arrived from a document, not from recollection.
- Verify each input against its original source document, not memory.
- Re-verify Strike against a source document — ten percent of error there moves the headline by ≈207.4 %.
- Confirm the example values are plausible for your own situation before replacing them.
- Re-run the calculation with a ±10 % stress on the two most sensitive inputs.
- Run the ±10 % stress on the top two levers and check that your conclusion survives both.
Step by step: getting a trustworthy Ultima out of this tool
The tool does the arithmetic; the routine below is what makes the arithmetic trustworthy. Five steps, a minute total — and afterwards you will know not just what the number is, but which assumption produced it.
Step 1 — start from evidence. The inputs behind Ultima are underlying price ($), strike ($), time to expiry (years), risk-free rate (%), implied volatility (%), option type. Write them down with their source next to each one; if you cannot name where a figure came from, that is the input most likely to be wrong. Repeating a calculation with the same numbers proves only that the arithmetic is reliable — which is worth knowing.
Step 2 — baseline before exploration. Get one defensible Ultima per 1%³ vol (×100,000) on screen before you start nudging numbers. Without a baseline, every variation looks interesting and none of them means anything. The fastest sanity test is to guess the answer first and see how far the calculation disagrees.
Step 3 — change one input at a time. Nudge a single field by ten percent, watch the headline and its companions move, then undo it. Two or three fields will dominate — the sensitivity table above already ranks them for this engine. The test of a good estimate is not that it feels right but that it changes what you do.
Step 4 — compute a corridor, not a point. Three runs — worst, likely, best — turn Ultima from a snapshot into a range. Plan against the pessimistic end and treat the optimistic end as the upside story. If a result feels obviously right, that is the moment to check the inputs hardest.
Step 5 — write the one-line memo. “Ultima per 1%³ vol (×100,000) = 8.880 on this date, from these inputs.” That line is the entire deliverable; everything else on this page is how you earned the right to write it. Comparisons are cheap: run the identical calculation for the alternative and look at the gap.
Who actually uses Ultima — and when
Investing & Markets arguments end at arithmetic or they do not end. Ultima is the arithmetic: 8.880 on the example inputs, reproducible by anyone who types the same numbers. The inputs are the part of the work that nobody else can do for you.
Used properly this is a five-minute filter that saves weeks. Compute Ultima, reject what fails, and only then bring in the expensive analysis for the survivors. Half of a good calculation is knowing which input you would hate to be wrong about.
There is a personal-administration use people overlook: periodic self-checking. Ultima quietly drifts as life changes — rates move, bodies change, costs compound — and re-running it a few times a year catches drift while correction is still cheap. The point of transparency is that a reader can disagree with a specific line rather than with you.
The questions that come up most about Ultima
What affects Ultima the most?
Ultima is driven above all by Strike. Perturbing only that field by ten percent either way produced 27.300 against the 8.880 baseline — a ≈207.4 % response, while Underlying price manages ≈204.3 %.
Do underlying price and strike matter equally?
Ultima here is co-driven: Strike and Underlying price produced ≈207.4 % and ≈204.3 % responses, which is a tie for practical purposes. Everything below those two is a rounding error by comparison.
What is a typical Ultima?
On the worked example above Ultima comes out at 8.880. Treat that as one point on a distribution rather than a norm: the bands, the companion outputs and the sensitivity ranking tell you how wide the plausible range really is.
How do I calculate Ultima by hand?
Use the stated model — Ultima = ∂Volga/∂σ — third vol derivative (∂³V/∂σ³) — and substitute in the order the tool prints (it starts from Ultima = ∂Volga/∂σ per pt³ = 0.00008880 (×100,000 = 8.880)). On the example inputs the hand route lands on 8.880, exactly what the calculator shows. A spreadsheet reproduces it in two or three cells; doing it once by hand is what makes the automated answer trustworthy.
What happens to Ultima if strike changes by 10 %?
The engine was re-run with that field moved by ten percent while everything else stayed fixed: the ultima per 1%³ vol (×100,000) goes across 1.706 to 27.300, a swing of roughly 207.4 % around the 8.880 baseline. That ratio is the practical definition of how much this input matters.
Is the Ultima result of 8.880 good or bad?
Neither, on its own — the label “Volga flexes 8.880 per vol point (×100,000) — the curvature-of-curvature that only 10+ point vol regime shifts monetize” is the band this figure falls in for the example inputs, and your own inputs will place you elsewhere on the same ladder. Judge the number against your history and your alternatives, and remember that a ten-percent error in Strike alone moves it by ≈207.4 %.
What affects Ultima the most?
Sensitivity depends on where you start, but the inputs Underlying price ($) and Strike ($) typically move Ultima further than the rest because they enter the formula multiplicatively. If you can only measure two things well, measure those and hold the others at conservative defaults.
Is Ultima accurate?
The arithmetic is exact: Ultima = ∂Volga/∂σ — third vol derivative (∂³V/∂σ³) is evaluated with full floating-point precision and every substitution is printed in the worked example. What limits accuracy is input quality and the model’s assumptions, so treat the 8.880 result as the correct answer to the question you typed — and check the assumptions section before acting on it.
Does ultima appear in risk systems?
Some vol desks print the full vol-Greek stack (vega, volga, ultima, vanna); risk aggregation is dominated by the first two.
Is ultima how VVIX relates to options?
Loosely — VVIX prices vol-of-vol; ultima is the slice of it inside an individual contract’s vol profile.
Does this Ultima tool store my numbers?
No. The entire calculation runs in your browser session; nothing entered above is transmitted or saved, which is the correct behaviour for sensitive Investing & Markets figures and the reason you can re-run this page as often as you like.
What the labels on this page mean
- Volga after +10 pts (×10,000)
- 10.947 in the default Ultima calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Volga now (×10,000)
- 2.068 for the Ultima worked case; it sits next to the main result because it carries context the headline leaves out.
- Ultima ×100,000 per contract
- 888.0 in the default Ultima calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Ultima per 1%³ vol (×100,000)
- The primary output of this tool: 8.880 for the default inputs, computed exactly from Ultima = ∂Volga/∂σ — third vol derivative (∂³V/∂σ³).
- Worked example
- The step-by-step substitution printed above, so the arithmetic can be re-derived by hand or in a spreadsheet.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “Volga flexes 8.880 per vol point (×100,000) — the curvature-of-curvature that only 10+ point vol regime shifts monetize”, and your inputs may not.
- Sensitivity (±10 %)
- How far the Ultima moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
The provenance of this Ultima calculation
The model is published arithmetic implemented exactly as written: Ultima = ∂Volga/∂σ — third vol derivative (∂³V/∂σ³). Source of record: Jaubert et al., Advanced Greeks (2011). Where a professional version of this calculation differs, it differs in inputs and jurisdiction-specific coefficients, not in the algebra — which is why re-checking your inputs beats hunting for a second calculator.
Treat the output as a precise answer to the question you typed: exact arithmetic, transparent formula, published source, no hidden rounding. What that buys you is reproducibility — two people, two browsers, the same arithmetic, and a printed trail from inputs to result that either of them can audit.
Next steps after the calculation
Re-run the calculation with your own figures, stress the two most sensitive inputs by ±10 %, and only then act on the result.
The audience here is people who need one trustworthy number before their next decision. Everything you need to defend the result — formula, substitution, sensitivity, source — is printed above, so the only remaining variable is the honesty of your inputs.