# Gamma

**BSM-08 · Black–Scholes & Greeks · Core**

**Lesson question:** What does Gamma measure, in which units, and what does it not predict?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain Gamma in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.

**Prerequisites:** BSM-03

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.

## Calculation popup: Calculate Gamma

```text
Gamma=∂Delta/∂S=e^(−qT)phi(d1)/(S sigma sqrt(T))
```

T is years remaining; calendar time has the opposite derivative sign. N is the standard normal CDF and phi its density.

1. d1=0.070239316937; d2=-0.001433455448.

2. Raw=0.055478868929; display multiplier=1.

3. Displayed=0.055478868929.

4. Local affected measure=0.541434333635.

5. Recomputed affected measure=0.541397294812.

**Result:** 0.055478868929 Delta change per $1 spot change

## Graph: Gamma across controlled states

**Horizontal:** Spot (USD)  
**Vertical:** Delta change per $1 spot change

Original BSM calculation with only the x-axis input varied. All other inputs are stated in the example; the curve is not a prediction.

Complete numerical data are embedded in `scripts/black-scholes-greeks/BSM-08.js` and displayed as an accessible table in the HTML lesson.

## 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

### What is the displayed Gamma in this example?

**Answer:** 0.055478868929 Delta change per $1 spot change.

**Why:** Always state the unit.

### Does model agreement prove the same live-market change?

**Answer:** No.

**Why:** Actual inputs, instrument features and executable quotes can differ. **Critical risk check.**

## Takeaway

Translate Gamma into its specific input, affected quantity and unit before applying it. Revalue when the local approximation no longer fits.

## References and provenance

The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations.

- [OIC Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks)

- [OIC Gamma](https://www.optionseducation.org/advancedconcepts/gamma)


## Plain-English quick read

### Plain English

Gamma measures how unstable your Delta is.

### 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.


## Try it in the working calculator

1. Plot Gamma against spot at 30 days.
2. Reload with seven days and compare the concentration near the strike.
3. Explain why a positive Gamma is not itself a directional forecast.

The working lab uses fictional shared inputs; the printed worked example above stays fixed. Open it from the lesson's same-page Live Greeks Lab control. The engine prices European calls and puts with continuous yield and calendar days/365.