Plain English
Gamma measures how unstable your Delta is.
Black–Scholes & Greeks · BSM-08 · Core
Gamma
What does Gamma measure, in which units, and what does it not predict?
How to read 0.04
A $1 spot rise changes Delta by about +0.04 locally.
Typical sign
Long vanilla calls and puts have positive Gamma.
Where it matters
Near the money and near expiration.
What you will learn and prerequisites
- Explain Gamma in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
Read first: Read the notation
Gamma explains why Delta changes. A long standard European call and put have the same positive Gamma at matching inputs. That means Delta becomes more positive on a rise and more negative on a decline.
Positive Gamma describes curvature, not a prediction of upward direction.
Read the derivative as a sentence
For a small move, dDelta≈Gamma×dS. In a price expansion the contribution is one-half Gamma×dS², because the slope changes gradually along the move. Omitting the half overstates the quadratic term.
Gamma can concentrate near strike and expiry. Far-away options behave differently, so not every option's Gamma increases as time runs down.
Use it within its scope
Compare Delta-only with Delta-plus-Gamma repricing. Convexity has a purchase cost and often negative Theta; positive Gamma alone does not establish a profitable trade. A fixed-Gamma approximation can itself become stale as the curvature changes.
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 Gamma is 0.055478868929. The displayed convention gives 0.055478868929 Delta change per $1 spot change. Isolate spot +$0.25 while holding every other input fixed.
Starting Delta is 0.527564616403. The local estimate after the shock is 0.541434333635; a full-model recomputation gives 0.541397294812. Approximation minus recomputation is +0.000037038823. 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
Gamma across controlled states
Vertical: Delta change per $1 spot change
Horizontal: Spot (USD)
1. Fictional example
Read the graph as a data table
| Spot (USD) | Fictional example |
|---|---|
| 80 | 0.00067794 |
| 81 | 0.00111773 |
| 82 | 0.00177834 |
| 83 | 0.00273367 |
| 84 | 0.00406485 |
| 85 | 0.00585316 |
| 86 | 0.00817056 |
| 87 | 0.0110681 |
| 88 | 0.014564 |
| 89 | 0.0186331 |
| 90 | 0.0231996 |
| 91 | 0.0281348 |
| 92 | 0.0332611 |
| 93 | 0.0383627 |
| 94 | 0.0432012 |
| 95 | 0.0475356 |
| 96 | 0.0511432 |
| 97 | 0.0538396 |
| 98 | 0.0554941 |
| 99 | 0.0560403 |
| 100 | 0.0554789 |
| 101 | 0.0538748 |
| 102 | 0.0513479 |
| 103 | 0.0480592 |
| 104 | 0.0441954 |
| 105 | 0.0399528 |
| 106 | 0.0355222 |
| 107 | 0.0310772 |
| 108 | 0.0267655 |
| 109 | 0.0227034 |
| 110 | 0.0189747 |
| 111 | 0.0156319 |
| 112 | 0.012699 |
| 113 | 0.010177 |
| 114 | 0.00804868 |
| 115 | 0.0062841 |
| 116 | 0.00484537 |
| 117 | 0.00369083 |
| 118 | 0.00277829 |
| 119 | 0.0020674 |
| 120 | 0.00152125 |
A correct number can be misread
The stated unit is Delta 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 Gamma in this example?
Reveal the worked answer
0.055478868929 Delta 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 Gamma 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 at 30 days.
- Reload with seven days and compare the concentration near the strike.
- Explain why a positive Gamma is not itself a directional forecast.
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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