# Color

**BSM-19 · Black–Scholes & Greeks · Advanced**

**Lesson question:** What does Color 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 Color in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.

**Prerequisites:** BSM-07, BSM-08, BSM-10

Color measures how Gamma changes as calendar time passes. It is sometimes called Gamma decay, but decay is a nickname rather than a sign guarantee. Near strike, Gamma can increase as expiry approaches.

The derivative helps plan how tomorrow's hedge instability can differ from today's.

## Read the derivative as a sentence

This course uses calendar time, opposite to remaining-maturity differentiation. Divide the annual quantity by 365. Matching call and put Color agree because their Gamma agrees in this model.

A daily estimate on a contract with hours left needs an accepted horizon and boundary handling. It cannot be extrapolated past expiry.

## Use it within its scope

Use Color to identify where hedge intensity may concentrate with time. Actual spot movement can carry the option away from the concentration, so an isolated clock experiment is not a complete path forecast.

## 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 Color is 0.339716300896. The displayed convention gives 0.000930729591 Gamma change per calendar day. Isolate one calendar day passing while holding every other input fixed.

Starting Gamma is 0.055478868929. The local estimate after the shock is 0.056409598521; a full-model recomputation gives 0.056433480449. Approximation minus recomputation is -0.000023881928. 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 Color

```text
Color=−∂Gamma/∂T=Gamma[q+d1 A+1/(2T)], where A=∂d1/∂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=0.339716300896; display multiplier=0.0027397260274.

3. Displayed=0.000930729591.

4. Local affected measure=0.056409598521.

5. Recomputed affected measure=0.056433480449.

**Result:** 0.000930729591 Gamma change per calendar day

## Graph: Color across controlled states

**Horizontal:** Calendar days remaining  
**Vertical:** Gamma change per calendar day

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-19.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is Gamma change per calendar day. 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 Color in this example?

**Answer:** 0.000930729591 Gamma change per calendar day.

**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 Color 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

### Question it answers

If spot stays still, how will my convexity change tomorrow?

### Why it matters

Gamma can change rapidly close to expiry.

### Use

Helpful for planning future hedge intensity.

### Naming warning

Higher-order Greek labels vary by platform.


## Try it in the working calculator

1. Record Gamma with 30 days and then 29 days remaining.
2. Compare that change with Color/day.
3. Distinguish a calendar-time derivative from a time-remaining derivative.

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.


### Formula helper

For the helper A in this written formula, A = ∂d₁/∂T = [(r−q+σ²/2)T−ln(S/K)]/(2σT√T). The live lab labels the same helper D to avoid confusion with a discounted-price term.