Speed Calculator
The formula
What the speed tells you
How the position’s convexity itself leans: after a rally, is your gamma thicker or thinner? Speed answers it, and it is the first term market makers price when hedging gaps — because gaps change gamma, not just delta.
Calculation: Speed = ∂Γ/∂S — how gamma changes with spot (third derivative). Enter your inputs and every substitution is shown below the result.
Reading the result
Short-dated near-money options carry high speed: a gap through the strike converts gamma violently — this is why pin risk and gap risk are the same conversation. Far from ATM, speed tells you convexity is dying.
Rules of thumb: Speed matters mostly inside ±5% of spot with <30 days left; beyond that it is rounding
Limitations and best practice
Speed is a third-order Greek: estimation noise from market prices (bid/ask and stale IV) often exceeds the signal. Use it on model grids, and treat chain-implied speed numbers as directional only.
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
What is “speed” vs “zomma”?
Why do gap traders care?
Is speed ever negative?
Can I see speed on ThinkorSwim?
Source & methodology
Third-order BS Greeks (Jaubert, Lozano & Talpondeau).
Speed: what this tool answers
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 a Speed 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.
In Investing & Markets the same quantity is often quoted under two or three different conventions. The formula printed above fixes one convention and applies it consistently, which is what makes results from this page comparable with each other.
Every investing metric in Investing & Markets is a bet about the future priced in today’s language. Speed translates that bet: Speed = ∂Γ/∂S — how gamma changes with spot (third derivative). The translation is mechanical and reliable; the judgement is deciding whether the inputs you feed it are honest forecasts or wishful numbers.
Nothing below is stock text pasted around a number: the worked example computes speed (∂Γ/∂S ×10,000) = -2.391 /$1 from this tool’s own defaults, and the sensitivity section ranks the inputs that move the result — Underlying price first, at ≈119.5 % of the headline for a ten-percent error either way in that input. Work down the page and by the end you could re-derive the result in a spreadsheet from memory.
Every input in Speed, and what it does to the answer
The engine implements a single stated model: Speed = ∂Γ/∂S — how gamma changes with spot (third derivative). There is no hidden coefficient and no login-gated precision — the formula printed is the formula executed, which is also what makes the result citable.
Underlying price ($) (example value 140) is a plain measured quantity — the unit shown in the label is the unit the formula expects. Use the figure exactly as the source document states it, not a rounded memory of it — an omitted line is the classic silent error.
Strike ($) (example value 105) is a plain measured quantity — the unit shown in the label is the unit the formula expects. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Time to expiry (years) (example value 1) is a plain measured quantity — the unit shown in the label is the unit the formula expects. Be precise about the boundary (exact date, start of month); time inputs decide the exponent in most formulas.
Risk-free rate (%) (example value 5) is a plain measured quantity — the unit shown in the label is the unit the formula expects. Enter it exactly as your document states it — quoted annual, monthly and effective rates are three different numbers.
Implied volatility (%) (example value 25) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Check the convention first: if the source says “6.5 % per year”, type 6.5 and let the formula do the rest.
Option type (example value Call) is a selector, not a measurement: the engine follows a different route for each option. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
The default Speed calculation, written out in full
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Underlying price ($), Strike ($), Time to expiry (years), Risk-free rate (%). The endpoint of that chain is -2.391 /$1 of speed (∂Γ/∂S ×10,000), sitting in the “A $3 underlying move reshapes gamma by -0.00072 (25% of current gamma) — convexity itself is directional” band.
- Speed = −Γ(1/S + d1/(Sσ√T)) = -0.0002391
- Speed flips sign across the strike — it is the skew of convexity, not its level
- +2%/−2% spot scenarios leave gamma at 0.0023 / 0.0036
Γ after +2% move — 0.002. If your inputs change, this is usually the first companion figure to move.
Γ after −2% move — 0.004. This is the number experienced users quote when the headline alone would mislead.
Speed asymmetry ×10,000 — -13.388 gamma swing over 4%. If your inputs change, this is usually the first companion figure to move.
Current gamma — 0.003. Read next to the headline, this is the figure that shows the shape of the trade-off.
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 Speed 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?
| Underlying price setting | Speed (∂Γ/∂S ×10,000) |
|---|---|
| 80 % of the example value | -6.537 /$1 |
| 90 % of the example value | -5.247 /$1 |
| example inputs | -2.391 /$1 |
| 110 % of the example value | -0.799 /$1 |
| 120 % of the example value | -0.219 /$1 |
That spread — -6.537 /$1 at the low end, -0.219 /$1 at the high — is what a careful read of Speed looks like: a band you can state out loud, with the reason for each edge. Reproduce it in the form above by editing Underlying price alone; the substitution steps will follow your figures exactly.
What actually moves your Speed
Generic advice says “it depends on your inputs”; this section says exactly how much. Each field was perturbed by ±10 % in isolation and the speed (∂Γ/∂S ×10,000) recomputed, so the ranking below is measured rather than asserted. It is the fastest honest answer to “what matters most in Speed?” — and it is specific to these defaults, because sensitivity moves with the starting point.
- Underlying price +10 % → Speed = -0.799 /$1 · −10 % → -5.247 /$1 (the speed (∂Γ/∂S ×10,000) shifts ≈119.5 %)
- Strike +10 % → Speed = -4.109 /$1 · −10 % → -0.861 /$1 (the speed (∂Γ/∂S ×10,000) shifts ≈71.9 %)
- Implied volatility: ten percent up gives -2.413 /$1, ten percent down gives -2.226 /$1 — ≈6.9 % of the speed (∂Γ/∂S ×10,000) either way
- Risk-free rate +10 % → Speed = -2.348 /$1 · −10 % → -2.434 /$1 (the speed (∂Γ/∂S ×10,000) shifts ≈1.8 %)
- Time to expiry +10 % → Speed = -2.377 /$1 · −10 % → -2.375 /$1 (the speed (∂Γ/∂S ×10,000) shifts ≈0.6 %)
Underlying price is the dominant lever here: ten percent either way moves the result by about 119.5 %, from -5.247 /$1 to -0.799 /$1. Measure that input from a document rather than from memory, and treat Strike as the second priority (≈71.9 % per ten percent). The remaining fields can sit at safe defaults without materially changing the answer — knowing which is which is what separates a careful calculation from an anxious one.
Interpreting the Speed verdict
A number on its own is trivia; a number against a range is intelligence. The default case lands in the “A $3 underlying move reshapes gamma by -0.00072 (25% of current gamma) — convexity itself is directional” band, and bands are deliberately wide because real situations overlap. How much of that band is luck? On Underlying price alone ten percent either way covers -5.247 /$1 to -0.799 /$1. The practical reading rules: compare against your own history before comparing against other people; treat band boundaries as zones, not walls — one point on the wrong side of a threshold rarely changes a real outcome; and when a companion output contradicts the headline, believe the pair.
Where Speed sits next to similar measures: this tool answers “how much is the Speed”, 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.
The errors that quietly break a Speed figure
Speed is a third-order Greek: estimation noise from market prices (bid/ask and stale IV) often exceeds the signal. Use it on model grids, and treat chain-implied speed numbers as directional only. 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.
Reading Speed as a promise rather than a conditional. Every number here is of the form “if these inputs, then this result”; when an input moves, the honest response is to recompute rather than to defend the old figure.
This model has its own particular trap: Underlying price carries ≈119.5 % of the response per ten percent of error, so effort spent elsewhere is largely decorative.
A final trap is presentation: reporting Speed to more decimal places than the inputs support. The tool prints what the arithmetic gives; you should quote only what your least precise field justifies.
Before you trust your Speed: a short checklist
- Record the date and the inputs alongside the number so future-you knows what was assumed.
- Compare the result with one independent method or published benchmark.
- Check Underlying price before anything else: it is the measured dominant input, ≈119.5 % of Speed per ten percent.
- Confirm the example values are plausible for your own situation before replacing them.
- Compare the output against one published benchmark or a colleague’s independent estimate.
- Confirm the answer against one independent method or published benchmark for Speed.
A routine for decision-grade Speed
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 — one document, one pass. Open the statement, label or record that holds underlying price ($), strike ($), time to expiry (years), risk-free rate (%), implied volatility (%), option type and transcribe straight into the form. Transcription from a single source removes the mismatch errors that appear when figures are assembled from three different places. Say the answer out loud before you save it; if it sounds wrong, the model did not change, your understanding did.
Step 2 — start from the case you can defend. Enter pessimistic-but-honest figures first; the optimistic run is a ceiling, not a plan. A conclusion that holds on conservative inputs holds everywhere. The test of a good estimate is not that it feels right but that it changes what you do.
Step 3 — play with the inputs on purpose. Ten percent up, ten percent down, one field at a time. Within a few minutes you will know this calculation better than most people who quote its results. Most surprises in numbers are surprises about inputs, not about formulas.
Step 4 — stress before you commit. A decision that survives a ten-percent squeeze on its two biggest inputs needs no luck to be sound. One that does not survive is a decision about hope, not about arithmetic. The input you reach for first is usually the one you measured longest ago.
Step 5 — save the story with the figure. A number without its inputs and its date is unusable within a quarter. Record -2.391 /$1, the assumptions behind it, and today’s date; your future self will be auditing a stranger’s arithmetic. Most surprises in numbers are surprises about inputs, not about formulas.
Where Speed matters in practice
Investing & Markets arguments end at arithmetic or they do not end. Speed is the arithmetic: -2.391 /$1 on the example inputs, reproducible by anyone who types the same numbers. Compare the calculation to the rough guess you arrived with, and ask why they differ.
The companion outputs — Γ after +2% move and Γ after −2% move — exist for the second stage of that screen: the headline sizes the prize, while the companions describe the trade-off you are accepting to get it. A figure nobody can reproduce is a figure nobody will act on twice.
Treat Speed like a smoke alarm rather than a painting — its value is in being tested periodically, not in being admired once. A calculation you cannot explain to somebody else is a calculation you cannot defend.
The questions that come up most about Speed
What affects Speed the most?
Speed is driven above all by Underlying price. Perturbing only that field by ten percent either way produced -0.799 /$1 against the -2.391 /$1 baseline — a ≈119.5 % response, while Strike manages ≈71.9 %.
Which matters more: strike or underlying price?
Underlying price, by a factor of about 1.7. Concretely: ten percent on Underlying price is worth ≈119.5 % of Speed, the same nudge on Strike only ≈71.9 %. That ranking is re-computed for each page from that page’s own defaults.
What is a typical Speed?
With the example inputs on this page the result is -2.391 /$1, labelled “A $3 underlying move reshapes gamma by -0.00072 (25% of current gamma) — convexity itself is directional”. “Typical” varies with situation, which is why every result here ships with its verdict band and its sensitivity list: they turn one figure into a range you can locate yourself inside.
How do I calculate Speed by hand?
Use the stated model — Speed = ∂Γ/∂S — how gamma changes with spot (third derivative) — and substitute in the order the tool prints (it starts from Speed = −Γ(1/S + d1/(Sσ√T)) = -0.0002391). On the example inputs the hand route lands on -2.391 /$1, 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 Speed if underlying price changes by 10 %?
On these inputs a ±10 % change in underlying price takes Speed to -0.799 /$1 against the -2.391 /$1 example — around 119.5 % of the headline. The same test on every other field is in the sensitivity section.
Is the Speed result of -2.391 /$1 good or bad?
The tool answers “what”, not “whether”: -2.391 /$1 carries the verdict “A $3 underlying move reshapes gamma by -0.00072 (25% of current gamma) — convexity itself is directional” for the default case, which is a starting interpretation rather than a verdict on your situation. Re-run with your own figures and compare the bands.
How do I calculate Speed manually?
Write down the inputs, then follow the printed steps in order — Speed = ∂Γ/∂S — how gamma changes with spot (third derivative). A spreadsheet replicates it with two or three cells. The manual route is worth doing once to build intuition; after that the calculator saves time and eliminates transcription slips.
What is a good Speed?
Markets give away free benchmarks: anything that beats the relevant index after costs earns the label "good", and this tool’s verdict band translates Speed against experience ranges. Before that, check that the inputs are realistic — a great Speed built on optimistic inputs is just a nicer-looking error.
What is “speed” vs “zomma”?
Speed is the spot derivative of gamma (∂Γ/∂S); zomma is the vol derivative (∂Γ/∂σ) — same curvature, different axis.
Why do gap traders care?
A gap skips delta hedging entirely; the P&L is gamma plus speed — knowing how gamma re-sits is the difference between pin and wound.
Is my data private on this page?
Yes — computation happens locally in your browser, with no account, no tracking of your inputs and nothing sent to a server. The printed formula is the whole story of what happens to your figures.
The vocabulary of Speed
- Γ after +2% move
- 0.002 in the default Speed calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Γ after −2% move
- 0.004 on the example inputs — a companion figure printed beside the headline of Speed so that no result is read in isolation.
- Speed asymmetry ×10,000
- -13.388 gamma swing over 4% in the default Speed calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Speed (∂Γ/∂S ×10,000)
- The primary output of this tool: -2.391 /$1 for the default inputs, computed exactly from Speed = ∂Γ/∂S — how gamma changes with spot (third derivative).
- Worked example
- Every substitution in order, from the example inputs to -2.391 /$1 — the audit trail for the headline figure.
- Verdict band
- The plain-language label — here “A $3 underlying move reshapes gamma by -0.00072 (25% of current gamma) — convexity itself is directional” — mapping the computed figure onto experience-based ranges for Investing & Markets.
- Sensitivity (±10 %)
- How far the Speed moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Method, sources and what this tool cannot tell you
Everything above runs on one stated formula, applied without hidden coefficients: Speed = ∂Γ/∂S — how gamma changes with spot (third derivative). Source of record: Third-order BS Greeks (Jaubert, Lozano & Talpondeau). Specialist software layers extra coefficients on top of the same algebra; none are hidden here, and that transparency is what makes a disagreement traceable to a specific input instead of a black box.
Treat the output as a precise answer to the question you typed: exact arithmetic, transparent formula, published source, no hidden rounding. Nothing is smoothed, weighted or adjusted behind the scenes; where judgement enters, this page says so and names the input responsible.
What to do with your Speed next
Save the number together with its inputs and today’s date; a Speed figure without context is unusable the moment an input changes.
Whoever you are (people who need one trustworthy number before their next decision), the next step is the same: replace the example inputs with your own, note the date, and re-run whenever a figure moves. In Investing & Markets, a dated figure is a trend, and an undated one is a rumour.