# Read a Greek curve

**BSM-05 · Black–Scholes & Greeks · Core**

**Lesson question:** What does the shape add to a single number?

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

## By the end

- Explain the mechanism in the lesson question.
- Reproduce the worked comparison and recognize its failure case.

**Prerequisites:** BSM-01

A curve shows how a sensitivity changes across controlled states. Its horizontal axis is not automatically time; a spot scan contains many hypothetical spot values at the same maturity.

Read axes, fixed inputs and units before assigning an interpretation to the shape.

## Understand the mechanism

On a price-versus-spot curve the slope is Delta and the curvature Gamma. On a Delta-versus-spot curve, slope is Gamma. A graph of Vega versus maturity answers a different question again.

Scaling can hide or exaggerate apparent changes. The plotted current marker must represent the actual current input, not a clamped substitute at the chart boundary.

## Walk through the process

Choose one varying input and name everything held fixed. Compare the current state with a nearby point, then a larger move. Explain why local approximations degrade.

Keep a table equivalent so exact points and units remain readable without relying on the graphic alone.

## Read before applying

All numerical market states are hypothetical. The calculations illustrate a European continuous-yield model, not a quote feed. Financial-model validation, instrument suitability and executable pricing are separate questions.

## A worked example

At spot 100 the shared model Delta is 0.527565. At 101 it is 0.582324. Their difference demonstrates that the hedge ratio changes with spot.

The graph scans spot 80–120 at a fixed thirty-day maturity. It is a set of counterfactual states, not a forecast that price travels from left to right. No live chart data are used.

## Graph: Read a Greek curve—worked comparison

**Horizontal:** Hypothetical spot (USD)  
**Vertical:** Delta (per unit)

Fictional data from the written example. The chart organizes the comparison; its shape is not a forecast.

Complete numerical data are embedded in `scripts/black-scholes-greeks/BSM-05.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

A reader treats a scenario scan as a future trajectory or compares axes with different units as though their slopes were identical.

## Check your understanding

### Is the x-axis here a forecast timeline?

**Answer:** No; it is hypothetical spot at fixed maturity.

**Why:** The varied input defines the experiment.

### What is the critical failure to avoid?

**Answer:** A reader treats a scenario scan as a future trajectory or compares axes with different units as though their slopes were identical.

**Why:** Explain this risk before treating the example as a decision rule. A correct calculation does not guarantee a favorable market outcome. **Critical risk check.**

## Takeaway

Return to the question: What does the shape add to a single number? State the observed facts, the assumptions and the failure condition separately.

## 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)


## Plain-English quick read

### Change one input

Choose spot, volatility or time remaining for the horizontal axis. All other market inputs stay fixed.

### Ask about the shape

Price's local slope is Delta. Delta's local spot slope is Gamma. A single number becomes easier to interpret when you see its neighborhood.

### Use the actual state

The marked state is your entered value. The chart domain expands to include it instead of clamping and mislabeling another point.

### Time-axis caution

Moving right on a days-remaining chart means a longer option life. It is not a picture of calendar time passing.


## Try it in the working calculator

1. In Greek curves plot Delta against spot.
2. Move to Gamma using the same state; compare where each changes most.
3. Set spot well outside the old default range; confirm the marked input is still the actual value.

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.