# Vomma / Volga

**BSM-15 · Black–Scholes & Greeks · Advanced**

**Lesson question:** What does Vomma / Volga 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 Vomma / Volga in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.

**Prerequisites:** BSM-07, BSM-08, BSM-10

Vomma measures how Vega changes as volatility changes. It is also called Volga. It adds the curvature missing from a constant-Vega estimate.

A long option need not have positive Vomma everywhere. Its sign in this model depends on d1×d2.

## Read the derivative as a sentence

Two volatility-point conversions produce the /10,000 display: the affected Vega is already per point, and the new IV move is also measured in points. Raw Vomma times 0.01 instead estimates change in raw Vega.

For price P/L, the quadratic term is one-half Vomma_raw×dSigma². Keep the half and unit conversions separate.

## Use it within its scope

Compare small and larger shocks rather than assigning permanent convexity from a strategy name. Near some ATM states the derivative is negative. Paying for long volatility still requires a view on realized movement, IV repricing and costs.

## 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 Vomma / Volga is -0.004591139275. The displayed convention gives -0.000000459114 Vega-per-point change per volatility point. Isolate IV +1 volatility point while holding every other input fixed.

Starting Vega per volatility point is 0.113997675882. The local estimate after the shock is 0.113997216768; a full-model recomputation gives 0.113996792281. Approximation minus recomputation is +0.000000424487. 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 Vomma / Volga

```text
Vomma_raw=∂²V/∂sigma²=Vega_raw d1 d2/sigma; displayed=raw/10,000
```

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=-0.004591139275; display multiplier=0.0001.

3. Displayed=-0.000000459114.

4. Local affected measure=0.113997216768.

5. Recomputed affected measure=0.113996792281.

**Result:** -0.000000459114 Vega-per-point change per volatility point

## Graph: Vomma / Volga across controlled states

**Horizontal:** IV (%)  
**Vertical:** Vega-per-point 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-15.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is Vega-per-point 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 Vomma / Volga in this example?

**Answer:** -0.000000459114 Vega-per-point 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 Vomma / Volga 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

### Question it answers

If vol moves again, will my Vega be larger or smaller?

### Relationship

Vega is the first derivative; Vomma is the next layer.

### Sign

Depends on moneyness and the market state.

### Use

Important for larger volatility shocks.


## Try it in the working calculator

1. Record Vega per point; raise IV by one point and compare.
2. Read Vomma's unit as the change in Vega-per-point per volatility point.
3. Do not confuse that derivative with the ½-weighted curvature contribution to price.

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.