# Vanna

**BSM-14 · Black–Scholes & Greeks · Advanced**

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

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

Vanna links directional and volatility risk. It tells how Delta changes with IV, or equivalently how raw Vega changes with spot in a smooth model. A hedge can therefore drift even if spot stays quiet while volatility changes.

Its sign depends on the state, not simply call versus put.

## Read the derivative as a sentence

Matching European calls and puts have the same Vanna here because their Delta difference does not depend on sigma. A per-vol-point display uses a one-point shock as one; raw Vanna uses 0.01.

In a second-order price expansion, the combined mixed term is Vanna×dS×dSigma, without another half after the two symmetric cross derivatives are combined.

## Use it within its scope

For dealer flow, net the signed book and reverse the option-Delta change to obtain the neutralizing hedge. Do not claim every IV decline forces the same direction. Surface shifts can be nonparallel and defeat a one-number shock assumption.

## 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 Vanna is 0.002279953518. The displayed convention gives 0.000022799535 Delta change per volatility point. Isolate IV +1 volatility point while holding every other input fixed.

Starting Delta is 0.527564616403. The local estimate after the shock is 0.527587415938; a full-model recomputation gives 0.527608461493. Approximation minus recomputation is -0.000021045555. 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 Vanna

```text
Vanna=∂Delta/∂sigma=∂Vega_raw/∂S=−e^(−qT)phi(d1)d2/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=0.002279953518; display multiplier=0.01.

3. Displayed=0.000022799535.

4. Local affected measure=0.527587415938.

5. Recomputed affected measure=0.527608461493.

**Result:** 0.000022799535 Delta change per volatility point

## Graph: Vanna across controlled states

**Horizontal:** IV (%)  
**Vertical:** Delta 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-14.js` and displayed as an accessible table in the HTML lesson.

## A correct number can be misread

The stated unit is Delta 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 Vanna in this example?

**Answer:** 0.000022799535 Delta 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 Vanna 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 implied volatility changes, how does my directional exposure move?

### Mixed derivative

It links spot risk and volatility risk.

### Sign

State-dependent; do not memorize a universal sign.

### Use

Useful when spot and implied volatility move together.


## Try it in the working calculator

1. Record Delta, then raise IV from 25% to 26%.
2. Compare that Delta change with Vanna per point.
3. Return to the starting state; inspect the Vanna cross-term only when both spot and IV shocks are nonzero.

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.