# Vega

**BSM-10 · Black–Scholes & Greeks · Core**

**Lesson question:** What does Vega measure, in which units, and what does it not predict?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain Vega in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.

**Prerequisites:** BSM-03

Vega measures sensitivity to the volatility input. Moving from 25% to 26% is one volatility point, or 0.01 in decimal sigma. It is not the same as increasing 25% by one percent relative, which gives 25.25%.

Vega is market terminology, not the name of a Greek letter.

## Read the derivative as a sentence

Raw Vega differentiates with respect to 1.00 decimal volatility. Dividing by 100 produces the per-point display. Matching standard European calls and puts have equal positive Vega.

Implied volatility is inferred from an option price under a model. It does not guarantee future realized movement. A sensitivity number says nothing by itself about whether volatility is cheap.

## Use it within its scope

A one-point shock is a useful local experiment. Larger shocks change Vega through Vomma and can interact with spot. Full repricing is preferable to extrapolating one Vega over an enormous move. A directional winner can still lose from an IV decline.

## The model boundary

The smooth formulas require positive spot, strike, volatility and remaining time. Zero time and zero volatility still admit explicit value limits, but smooth Greek availability needs separate rules. No invalid or zero input is silently replaced with a default. Continuous dividends, constant parameters and European exercise are modeling assumptions, not claims about every real option.

## One complete numerical experiment

The shared fictional European call uses spot 100, strike 100, 30 calendar days, sigma 25%, continuous rate 4% and yield 1%. Its BSM value is 2.977574155; d1=0.070239317 and d2=-0.001433455. Values are per option unit before contract multiplication.

Raw Vega is 11.399767588205. The displayed convention gives 0.113997675882 option-value change per volatility point. Isolate IV +1 volatility point while holding every other input fixed.

Starting option value is 2.977574155222. The local estimate after the shock is 3.091571831104; a full-model recomputation gives 3.091571458721. Approximation minus recomputation is +0.000000372384. The difference shows the limitation of a local derivative, not a claim that the market must follow either number.

The graph varies only its labeled horizontal input. It is a series of controlled states, not historical observations or a future trajectory. Read its units before interpreting sign or magnitude.

## Calculation popup: Calculate Vega

```text
Vega_raw=∂V/∂sigma=S e^(−qT)phi(d1)sqrt(T); Vega_per_point=raw/100
```

T is years remaining; calendar time has the opposite derivative sign. N is the standard normal CDF and phi its density.

1. d1=0.070239316937; d2=-0.001433455448.

2. Raw=11.399767588205; display multiplier=0.01.

3. Displayed=0.113997675882.

4. Local affected measure=3.091571831104.

5. Recomputed affected measure=3.091571458721.

**Result:** 0.113997675882 option-value change per volatility point

## Graph: Vega across controlled states

**Horizontal:** IV (%)  
**Vertical:** option-value change per volatility point

Original BSM calculation with only the x-axis input varied. All other inputs are stated in the example; the curve is not a prediction.

Complete numerical data are embedded in `scripts/black-scholes-greeks/BSM-10.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is option-value change per volatility point. Mixing raw decimal derivatives with point shocks or calendar time with remaining time can magnify or reverse the result.

The controlled comparison also omits live spread, surface dynamics, unsupported exercise effects and execution. A local sensitivity is not a market forecast or a guarantee of realized P/L.

## Check your understanding

### What is the displayed Vega in this example?

**Answer:** 0.113997675882 option-value change per volatility point.

**Why:** Always state the unit.

### Does model agreement prove the same live-market change?

**Answer:** No.

**Why:** Actual inputs, instrument features and executable quotes can differ. **Critical risk check.**

## Takeaway

Translate Vega into its specific input, affected quantity and unit before applying it. Revalue when the local approximation no longer fits.

## 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)

- [OIC Vega](https://www.optionseducation.org/advancedconcepts/vega)


## Plain-English quick read

### Plain English

Vega is your exposure to the market’s volatility assumption.

### How to read 0.12

A one-volatility-point rise, such as 25% to 26%, adds about $0.12.

### Typical sign

Long vanilla calls and puts have positive Vega.

### Where it matters

Longer-dated and near-the-money options often carry more Vega.


## Try it in the working calculator

1. Keep spot and time fixed; apply +1 IV point.
2. Read Vega/vol point and compare its estimated change with repricing.
3. Repeat with +10 points to see why constant Vega is only an approximation.

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.