Read a derivative aloud
The symbol ∂ means a partial derivative: change one input while holding the others fixed. ∂V/∂S reads as the local change in option value when spot changes.
Black–Scholes & Greeks · BSM-03 · Core
Read the notation
What does each symbol stand for?
Spot and strike are different
Spot S is today's underlying price. Strike K is a contract term. Moving one while holding the other fixed is a different experiment from moving both.
Time has a direction
T means years remaining. One calendar day passing reduces T by 1/365. Theta, Charm, Color and Veta here use calendar-time derivatives.
Decimals versus points
The form accepts 25 for 25% IV, but the model uses 0.25. A change from 25% to 26% is +0.01 in the model.
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
S is spot, K strike, T years remaining, sigma annualized decimal volatility, r a continuous annual rate and q a continuous annual yield. V is model option value. N is the standard normal cumulative distribution and phi its density.
The symbols are shorthand for inputs you can explain aloud. A numerical formula is only meaningful when their units and definitions agree.
Understand the mechanism
d1=[ln(S/K)+(r−q+sigma²/2)T]/[sigma sqrt(T)] and d2=d1−sigma sqrt(T). A call is S e^(−qT)N(d1)−K e^(−rT)N(d2). A put is K e^(−rT)N(−d2)−S e^(−qT)N(−d1).
Calendar time advances while T decreases. Percentages entered as decimals matter: 25% is 0.25, not 25.
Walk through the process
Translate days to the stated year convention and check the instrument. This teaching model uses European exercise and continuous yield. Distinguish raw derivatives from displayed per-point or per-day values.
At T=0 calculate intrinsic explicitly. At sigma=0 use a deterministic discounted value limit; do not let division by zero silently substitute ordinary defaults.
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 S=K=100, T=30/365, sigma=0.25, r=0.04 and q=0.01, d1=0.070239317, d2=-0.001433455, and call value=2.977574155.
A 100-unit contract multiplies the value by 100 before fees. The chart varies strike, leaving the declared spot, time and other inputs unchanged. It is not a comparison of live contracts.
Open the calculation without leaving this lesson
Read the notation—worked comparison
Vertical: Call model value per unit (USD)
Horizontal: Strike (USD)
1. Fictional example
Read the graph as a data table
| Strike (USD) | Fictional example |
|---|---|
| 90 | 10.41 |
| 92 | 8.6063 |
| 94 | 6.92549 |
| 96 | 5.407 |
| 98 | 4.08414 |
| 100 | 2.97757 |
| 102 | 2.09151 |
| 104 | 1.41367 |
| 106 | 0.918769 |
| 108 | 0.573995 |
| 110 | 0.344742 |
What would invalidate the reading?
Entering 25 instead of 0.25 or changing days without the year conversion creates a different model. Rounding intermediate terms can distort small option values.
Check your understanding
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
What decimal represents annual volatility of 25%?
Reveal the worked answer
0.25.
The input unit controls every formula.
What is the critical failure to avoid?
Reveal the worked answer
Entering 25 instead of 0.25 or changing days without the year conversion creates a different model. Rounding intermediate terms can distort small option values.
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: What does each symbol stand for? State the observed facts, the assumptions and the failure condition separately.
Try it in the working lab
- Load Lesson example and open Formula steps.
- Check that 30 days becomes 30/365 years and 25% becomes 0.25.
- Locate N(d₁) and distinguish cumulative probability from φ(d₁), density height.
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.
Read this lesson as Markdown · Download its JavaScript content module