# Greeks comprehension review

**BSM-23 · Black–Scholes & Greeks · Core**

**Lesson question:** Can you explain the number and its boundary?

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

This review tests interpretation rather than memorization. Work the calculation before revealing its answer. A correct answer should include the unit and what the result does not establish.

Passing a local reading check is not certification of trading suitability, numerical implementation or a live model.

## Understand the mechanism

Distinguish a derivative from a finite forecast, decimal IV from vol points, calendar time from maturity and per-unit value from a full contract. Explain why a smooth formula can need a different boundary policy at expiry.

A saved visit or previous answer is not evidence that a learner understands a revised lesson.

## Walk through the process

Answer the checks, then describe an adverse interpretation error. Return to the specific lesson whose unit or mechanism remains unclear. There is no need to memorize all fifteen Greek names before applying the primary concepts carefully.

For a build, completion should depend on current-version required responses, not a generic progress percentage.

## 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

Suppose a standard contract has per-unit Delta 0.60, Gamma 0.04, Vega per point 0.12 and daily Theta −0.08. A small +0.50 spot shock gives approximately 0.30+0.005=0.305 price change per unit with Delta and Gamma. One vol point adds about 0.12; one day alone subtracts about 0.08.

For 100 units these isolated changes are 30.50, 12 and −8 dollars. They are controlled examples, not assured simultaneous realized P/L. The graph compares them in the common cash unit.

## Graph: Greeks comprehension review—worked comparison

**Horizontal:** Spot / IV / time isolated experiment  
**Vertical:** Approximate 100-unit P/L (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-23.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

Getting arithmetic right without matching units can still produce a wrong trade interpretation. A quiz result does not remove leverage, model or execution risk.

## Check your understanding

### What is Delta-plus-Gamma for dS=.50 with Delta=.60 and Gamma=.04?

**Answer:** 0.305 per unit, or 30.50 per 100-unit contract.

**Why:** .60×.50+.5×.04×.50².

### What is the critical failure to avoid?

**Answer:** Getting arithmetic right without matching units can still produce a wrong trade interpretation. A quiz result does not remove leverage, model or execution risk.

**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.**

### A move from 25% to 26% changes decimal sigma by what amount?

**Answer:** 0.01.

**Why:** It is one volatility point, not one percent relative.

### Does exact expiry use an ordinary smooth Gamma at strike?

**Answer:** No.

**Why:** Intrinsic payoff has a kink and needs explicit derivative availability handling. **Critical risk check.**

### May a zero IV input silently become the default IV?

**Answer:** No.

**Why:** Zero and missing are different inputs; use an explicit value limit. **Critical risk check.**

## Takeaway

Return to the question: Can you explain the number and its boundary? 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. Answer each practice question before revealing feedback.
2. For a missed unit question, reproduce the fixed example in the calculator.
3. Use personal review marks only for your reading record, not proof of trading readiness.

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.

## Additional practice quiz with feedback

### Call Delta is 0.60. What does it tell you?

1. The next stock move has a guaranteed 60% success rate.
2. A sufficiently small $1 spot rise initially changes option value by about +$0.60 per unit, other inputs fixed.
3. Tomorrow the option must rise $0.60.

**Answer:** A sufficiently small $1 spot rise initially changes option value by about +$0.60 per unit, other inputs fixed.

Delta is a local sensitivity. It is not a promise, realized P/L, or automatically a real-world probability.

### Why can a spot hedge become stale after price moves?

1. Gamma changes the option's Delta.
2. The contract multiplier automatically changes.
3. A put always becomes a call.

**Answer:** Gamma changes the option's Delta.

Gamma measures how Delta changes with spot; dynamic hedging also faces costs and other risks.

### Vega is 0.12 per volatility point. Which change matches one quoted unit?

1. 25% IV to 25.25% IV.
2. 25% IV to 26% IV.
3. 25% IV to 125% IV.

**Answer:** 25% IV to 26% IV.

A volatility point is a percentage point. A raw decimal-volatility derivative is divided by 100 for this quote.

### Which statement about long-option Theta is safest?

1. It is always negative under all inputs.
2. It is guaranteed daily cash income.
3. It is a local calendar-time sensitivity; it is often negative but exceptions exist.

**Answer:** It is a local calendar-time sensitivity; it is often negative but exceptions exist.

Moneyness, rates, yield and option type matter. Daily scaling is raw annual calendar-time Theta / 365 here.

### You enter seven days remaining and request thirty days passing. What should the lab do?

1. Use thirty days of Theta and silently reprice at expiry.
2. Reject the inconsistent scenario and ask for a valid elapsed time.
3. Replace seven days with thirty days.

**Answer:** Reject the inconsistent scenario and ask for a valid elapsed time.

Identical applied shocks are essential. Silent clipping in only one calculation invalidates the comparison.

### What happens when you set volatility to zero?

1. The application should restore its 25% default.
2. It should show a deterministic value limit and label unavailable smooth Greeks.
3. Every value must be zero.

**Answer:** It should show a deterministic value limit and label unavailable smooth Greeks.

Zero is a meaningful boundary input. Unavailable Greeks are not the same as zero Greeks.

### What does Vanna connect?

1. Spot/Delta and volatility/Vega.
2. Only rates and company earnings.
3. Share count and contract expiry.

**Answer:** Spot/Delta and volatility/Vega.

Vanna is a mixed spot–volatility derivative. It can affect hedging when volatility changes.

### Why might Lambda be unavailable for a tiny option value?

1. A tiny denominator makes percentage elasticity poorly conditioned under the stated display policy.
2. The option becomes free of risk.
3. All high-order Greeks are universally zero.

**Answer:** A tiny denominator makes percentage elasticity poorly conditioned under the stated display policy.

The lab requires unrounded V > 10^-8 × max(S,K). This is a transparent project convention, not an industry standard.

### Does a close match between approximation and model repricing validate a live trade?

1. Yes, it proves the quoted fill.
2. Yes, it removes early-exercise risk.
3. No. Both are model calculations, not executable quotes.

**Answer:** No. Both are model calculations, not executable quotes.

Spreads, costs, exercise features, surface changes and actual market inputs remain separate questions.