# Zomma

**BSM-18 · Black–Scholes & Greeks · Advanced**

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

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

Zomma connects spot curvature and volatility. It asks whether Gamma becomes larger or smaller after an IV change. That is a different question from Vega's change in option value.

This distinction prevents the blanket claim that lower IV reduces every type of option sensitivity.

## Read the derivative as a sentence

Near the ATM state used here, d1×d2−1 is negative, giving negative Zomma. Higher IV reduces the sharpness of the Delta transition; lower IV can increase it. Other states and signed portfolios differ.

The display is Gamma change per one volatility point. Raw and point-scaled derivatives must not be mixed.

## Use it within its scope

Stress spot and IV jointly when managing convexity. Recalculate after a major event rather than retaining the earlier Gamma profile. A small premium does not guarantee a small sensitivity of the hedge ratio.

## 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 Zomma is -0.221937819262. The displayed convention gives -0.002219378193 Gamma change per volatility point. Isolate IV +1 volatility point while holding every other input fixed.

Starting Gamma is 0.055478868929. The local estimate after the shock is 0.053259490737; a full-model recomputation gives 0.053344652798. Approximation minus recomputation is -0.000085162062. 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 Zomma

```text
Zomma=∂Gamma/∂sigma=Gamma(d1 d2−1)/sigma; displayed=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=-0.221937819262; display multiplier=0.01.

3. Displayed=-0.002219378193.

4. Local affected measure=0.053259490737.

5. Recomputed affected measure=0.053344652798.

**Result:** -0.002219378193 Gamma change per volatility point

## Graph: Zomma across controlled states

**Horizontal:** IV (%)  
**Vertical:** Gamma 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-18.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is Gamma 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 Zomma in this example?

**Answer:** -0.002219378193 Gamma 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 Zomma 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 Gamma](https://www.optionseducation.org/advancedconcepts/gamma)


## Plain-English quick read

### Question it answers

Will a volatility shock increase or reduce my spot convexity?

### Relationship

Gamma is not fixed when implied volatility changes.

### Use

Useful in convex books during volatility shocks.

### Sign

State-dependent.


## Try it in the working calculator

1. Record Gamma at 25% IV and then 26% IV.
2. Compare the change with Zomma/vol point.
3. Check why falling IV is not a universal reduction in every sensitivity.

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.