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.
Black–Scholes & Greeks · BSM-04 · Core
Walk through a calculation
How do you audit an option value instead of trusting a screen?
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.
Working calculator
The tool below is live. The written examples that follow retain their fixed fictional inputs.
What you will learn and prerequisites
- Explain the mechanism in the lesson question.
- Reproduce the worked comparison and recognize its failure case.
Read first: Greeks without intimidation
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.
Open the calculation without leaving this lesson
Walk through a calculation—worked comparison
Vertical: Call model value per unit (USD)
Horizontal: Hypothetical spot (USD)
1. Fictional example
Read the graph as a data table
| Hypothetical spot (USD) | Fictional example |
|---|---|
| 97 | 1.64581 |
| 98 | 2.03415 |
| 99 | 2.47789 |
| 100 | 2.97757 |
| 101 | 3.53265 |
| 102 | 4.14153 |
| 103 | 4.80169 |
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
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
Should a valid zero-volatility input become the default 25%?
Reveal the worked answer
No.
Use the explicit deterministic limit and appropriate Greek availability.
What is the critical failure to avoid?
Reveal the worked 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.
Explain this risk before treating the example as a decision rule. A correct calculation does not guarantee a favorable market outcome.
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.
Try it in the working lab
- Load Lesson example: call, S=100, K=100, 30 days, IV=25%, r=4%, q=1%.
- Verify value about 2.977574155 per unit before multiplier and costs.
- Change the type to put; compare both prices and return to call.
Your live inputs are independent of the fixed worked example above. Return to the “Lesson example” preset to reproduce it.
My reading record
Personal review marks are not quiz passes, course certification or trading approval. Old plugin progress is not overwritten.
Sources, assumptions and downloadable script
External references support definitions and conventions. Worked scenarios and prose are original. No current quotes or dealer inventory are claimed.
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