# Read the notation

**BSM-03 · Black–Scholes & Greeks · Core**

**Lesson question:** What does each symbol stand for?

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

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.

## 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: Read the notation—worked comparison

**Horizontal:** Strike (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-03.js` and displayed as an accessible table in the HTML lesson.

## 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

### What decimal represents annual volatility of 25%?

**Answer:** 0.25.

**Why:** The input unit controls every formula.

### What is the critical failure to avoid?

**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.

**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: What does each symbol stand for? 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

### 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.

### 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.


## Try it in the working calculator

1. Load Lesson example and open Formula steps.
2. Check that 30 days becomes 30/365 years and 25% becomes 0.25.
3. Locate N(d₁) and distinguish cumulative probability from φ(d₁), density height.

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.