# Ultima

**BSM-21 · Black–Scholes & Greeks · Advanced**

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

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

Ultima is the next volatility derivative after Vomma. It describes how volatility curvature changes again. Its purpose is understanding nonlinear exposure, not requiring a third-order calculation for every simple directional trade.

The sequence is Vega, changing Vega, then changing volatility curvature.

## Read the derivative as a sentence

Three point-scaling factors produce the displayed per-point-cubed convention. In a price expansion the third-order term is one-sixth Ultima_raw×dSigma³.

Nested finite differences can amplify cancellation and bump-size errors. An analytic formula or several converging bump choices is better evidence than one arbitrary nested difference.

## Use it within its scope

Large shocks need full repricing and a defensible volatility surface. Skew, term structure and exercise features may dominate model error before adding another Taylor term helps. High derivative order is not automatically high decision quality.

## 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 Ultima is -0.881871678693. The displayed convention gives -0.000000881872 Vomma-per-point² change per volatility point. Isolate IV +1 volatility point while holding every other input fixed.

Starting Vomma per volatility-point squared is -0.000000459114. The local estimate after the shock is -0.000001340986; a full-model recomputation gives -0.000001292425. Approximation minus recomputation is -0.000000048561. 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 Ultima

```text
Ultima=∂Vomma_raw/∂sigma=−Vega_raw/sigma²×[d1 d2(1−d1 d2)+d1²+d2²]; displayed=raw/1,000,000
```

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=-0.881871678693; display multiplier=1e-06.

3. Displayed=-0.000000881872.

4. Local affected measure=-0.000001340986.

5. Recomputed affected measure=-0.000001292425.

**Result:** -0.000000881872 Vomma-per-point² change per volatility point

## Graph: Ultima across controlled states

**Horizontal:** IV (%)  
**Vertical:** Vomma-per-point² 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-21.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is Vomma-per-point² 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 Ultima in this example?

**Answer:** -0.000000881872 Vomma-per-point² 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 Ultima 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

### Hierarchy

Vega → Vomma → Ultima.

### Question it answers

How stable is my volatility convexity during another vol move?

### Use

Mostly relevant in portfolios with nonlinear volatility exposure.

### Perspective

Usually less important than Delta, Gamma, Theta, and Vega for basic trading.


## Try it in the working calculator

1. Inspect Vomma and Ultima for the same fictional state.
2. Change volatility slightly, then substantially.
3. Explain why a local third derivative is not a complete shock model.

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.