Question it answers
How quickly will my Gamma estimate become stale after a spot move?
Black–Scholes & Greeks · BSM-17 · Advanced
Speed
What does Speed measure, in which units, and what does it not predict?
Relationship
Delta → Gamma → Speed.
Use
Useful around concentrated short-dated Gamma.
Interpretation
Sign indicates whether Gamma rises or falls as spot rises.
What you will learn and prerequisites
- Explain Speed in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
Speed asks how Gamma changes along the spot axis. Delta is slope, Gamma curvature and Speed the changing curvature. It explains why even a Gamma-aware approximation can become stale after a move.
Third order need not be mysterious: it is another comparison of an already familiar sensitivity.
Read the derivative as a sentence
For a small move, dGamma≈Speed×dS. In a third-order price expansion the term is one-sixth Speed×dS³, not Gamma's one-half.
Speed can have either sign. A positive-Gamma curve can rise on one side of its peak and fall on the other; positive curvature does not determine its slope.
Use it within its scope
Compare Gamma at nearby valid spots. If checking numerically, do not clamp one finite-difference endpoint and keep a symmetric denominator. Use valid symmetric points or an explicitly appropriate one-sided method and check bump convergence.
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 Speed is -0.001098481605. The displayed convention gives -0.001098481605 Gamma change per $1 spot change. Isolate spot +$0.25 while holding every other input fixed.
Starting Gamma is 0.055478868929. The local estimate after the shock is 0.055204248528; a full-model recomputation gives 0.055171778830. Approximation minus recomputation is +0.000032469698. 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
Speed across controlled states
Vertical: Gamma change per $1 spot change
Horizontal: Spot (USD)
1. Fictional example
Read the graph as a data table
| Spot (USD) | Fictional example |
|---|---|
| 80 | 0.000351331 |
| 81 | 0.000538725 |
| 82 | 0.000794868 |
| 83 | 0.00112944 |
| 84 | 0.00154662 |
| 85 | 0.0020422 |
| 86 | 0.0026013 |
| 87 | 0.00319698 |
| 88 | 0.00379076 |
| 89 | 0.00433487 |
| 90 | 0.00477658 |
| 91 | 0.00506399 |
| 92 | 0.00515244 |
| 93 | 0.0050107 |
| 94 | 0.00462578 |
| 95 | 0.00400555 |
| 96 | 0.0031787 |
| 97 | 0.0021921 |
| 98 | 0.0011058 |
| 99 | -1.33221e-05 |
| 100 | -0.00109848 |
| 101 | -0.00208938 |
| 102 | -0.00293736 |
| 103 | -0.00360869 |
| 104 | -0.00408594 |
| 105 | -0.00436734 |
| 106 | -0.00446474 |
| 107 | -0.00440045 |
| 108 | -0.00420361 |
| 109 | -0.00390664 |
| 110 | -0.00354202 |
| 111 | -0.00313981 |
| 112 | -0.00272589 |
| 113 | -0.00232105 |
| 114 | -0.00194064 |
| 115 | -0.0015949 |
| 116 | -0.00128955 |
| 117 | -0.0010266 |
| 118 | -0.000805235 |
| 119 | -0.000622703 |
| 120 | -0.000475035 |
A correct number can be misread
The stated unit is Gamma change per $1 spot change. 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 Speed in this example?
Reveal the worked answer
-0.001098481605 Gamma change per $1 spot change.
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 Speed 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
- Plot Gamma against spot, then plot Speed.
- Compare where Gamma increases and decreases.
- Explain why Gamma held constant after a large spot move becomes stale.
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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