# A usable Greek cheat sheet

**BSM-22 · Black–Scholes & Greeks · Core**

**Lesson question:** How do you turn the risk map into sentences?

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 cheat sheet should help read a number, not merely list symbols. For every Greek name the input, affected quantity, unit, sign convention and relevant limitation. Keep raw and market-scaled displays distinguishable.

Portfolio Greeks require signed quantities and multipliers before they can be combined.

## Understand the mechanism

Delta changes value with spot; Gamma changes Delta with spot; Theta changes value with calendar time; Vega changes value with IV; Rho and Epsilon change value with rates and yield. Lambda is percentage elasticity.

Vanna links spot and IV, Vomma changes Vega with IV, Charm changes Delta with time, Speed changes Gamma with spot, Zomma changes Gamma with IV, Color changes Gamma with time, Veta changes Vega with time and Ultima changes Vomma with IV.

## Walk through the process

Read the first four before adding higher derivatives. Use exact units and match dates across positions. Do not add the raw Greek numbers together: they have different dimensions.

For a portfolio comparison, translate a declared shock into dollar contributions first. Then aggregation becomes meaningful.

## 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

For the common call, spot +0.50, IV +1 point and one day elapsed give first terms 0.263782 Delta, 0.006935 Gamma, -0.051509 Theta and 0.113998 Vega per unit.

A 100-unit contract scales each by 100. These dollar contributions can be added; adding raw Delta to raw Vega cannot. The graph uses the shared shock and cash unit, not a misleading bar chart of unlike Greeks.

## Graph: A usable Greek cheat sheet—worked comparison

**Horizontal:** Delta / Gamma / Theta / Vega contribution  
**Vertical:** Model P/L per 100-unit contract (USD)

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-22.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

Blindly summing sensitivities with different units or opposite time conventions produces a meaningless total. Higher order does not automatically improve a mismatched model.

## Check your understanding

### What must happen before unlike Greeks become comparable P/L contributions?

**Answer:** Apply explicit matching shocks and contract scaling.

**Why:** Then the results share cash units.

### What is the critical failure to avoid?

**Answer:** Blindly summing sensitivities with different units or opposite time conventions produces a meaningless total. Higher order does not automatically improve a mismatched model.

**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: How do you turn the risk map into sentences? 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)

## Try it in the working calculator

1. Use the cheat sheet to name each input and unit.
2. Open the live lab to compare the six primary Greeks without adding unlike units.
3. Translate each into a common dollar contribution before combining them.

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.