# Walk through a calculation

**BSM-04 · Black–Scholes & Greeks · Core**

**Lesson question:** How do you audit an option value instead of trusting a screen?

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 transparent calculator exposes inputs, assumptions, intermediate terms and output units. The reader should be able to reproduce one example rather than treat a displayed price as authoritative.

Use the worksheet in this lesson as a controlled model experiment, not as a live quote or order recommendation.

## Understand the mechanism

Start with valid finite inputs. Spot and strike must be positive; remaining time and volatility may be zero, with explicit limits. Negative rates are not automatically invalid. A blank or malformed input is not the same as a legitimate zero.

The exact exercise style and dividend model must be declared. A European value alone does not validate an American equity option's fair value.

## Walk through the process

Compute d1 and d2, normal terms, discounted asset and strike contributions, then their difference. Check broad bounds and put-call parity at matching inputs. Finally scale per-unit output to contract quantity and multiplier.

A numerical reference check is different from testing a production application's input handling and presentation.

## 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

The shared call has d1=0.070239317 and d2=-0.001433455. Its model value is 2.977574155, or 297.7574 for one 100-unit contract before costs.

Now change only spot to 101: value becomes 3.532652359. Then reset and change only IV to 26%: value becomes 3.091571459. The graph repeats the spot experiment, so changes have a clear cause.

## Calculation popup: d1, d2 and price

```text
d1=[ln(S/K)+(r−q+sigma²/2)T]/(sigma sqrt(T)); d2=d1−sigma sqrt(T)
```

Use annual decimal inputs and T=days/365.

1. S=K=100; T=0.082191780822; sigma=.25; r=.04; q=.01.

2. d1=0.070239316937; d2=-0.001433455448.

3. Call=S exp(−qT)N(d1)−K exp(−rT)N(d2)=2.977574155222.

**Result:** 2.977574155 per unit.

## Graph: Walk through a calculation—worked comparison

**Horizontal:** Hypothetical spot (USD)  
**Vertical:** Call model value per unit (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-04.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

A calculator silently turns zero into a default, clamps a scenario differently in two panels, or labels an approximation exact. These are implementation errors, not harmless display choices.

## Check your understanding

### Should a valid zero-volatility input become the default 25%?

**Answer:** No.

**Why:** Use the explicit deterministic limit and appropriate Greek availability.

### What is the critical failure to avoid?

**Answer:** A calculator silently turns zero into a default, clamps a scenario differently in two panels, or labels an approximation exact. These are implementation errors, not harmless display choices.

**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 audit an option value instead of trusting a screen? 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

### Price first, explain second

Enter a fictional market state and read the per-unit model value. Then choose a Greek's calculation button to inspect d₁, d₂, the formula, scaling and interpretation.

### Check the contract multiplier

A per-unit value of 2.98 becomes about 298 for a 100-unit contract. The multiplier changes exposure, not the per-unit model price.

### Keep examples reproducible

The prose uses a fixed worked example. Your live inputs are saved separately, so experimentation cannot silently rewrite the numbers printed in the lesson.

### Limits are visible

Zero time means expiry. Zero volatility means a deterministic discounted payoff. Unavailable smooth Greeks are labeled, never replaced with defaults or zero.


## Try it in the working calculator

1. Load Lesson example: call, S=100, K=100, 30 days, IV=25%, r=4%, q=1%.
2. Verify value about 2.977574155 per unit before multiplier and costs.
3. Change the type to put; compare both prices and return to call.

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.