Question it answers
If implied volatility changes, how does my directional exposure move?
Black–Scholes & Greeks · BSM-14 · Advanced
Vanna
What does Vanna measure, in which units, and what does it not predict?
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.
What you will learn and prerequisites
- Explain Vanna in plain language.
- Reproduce the unit conversion.
- Compare a local estimate with model revaluation.
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.
Open the calculation without leaving this lesson
Vanna across controlled states
Vertical: Delta change per volatility point
Horizontal: IV (%)
1. Fictional example
Read the graph as a data table
| IV (%) | Fictional example |
|---|---|
| 10 | -0.00284263 |
| 12 | -0.00180229 |
| 14 | -0.00117387 |
| 16 | -0.000765559 |
| 18 | -0.000485429 |
| 20 | -0.000284965 |
| 22 | -0.000136605 |
| 24 | -2.37495e-05 |
| 26 | 6.4081e-05 |
| 28 | 0.000133767 |
| 30 | 0.000189977 |
| 32 | 0.000235968 |
| 34 | 0.000274072 |
| 36 | 0.000305988 |
| 38 | 0.000332983 |
| 40 | 0.000356014 |
| 42 | 0.000375818 |
| 44 | 0.000392967 |
| 46 | 0.000407911 |
| 48 | 0.000421009 |
| 50 | 0.000432549 |
| 52 | 0.000442767 |
| 54 | 0.000451853 |
| 56 | 0.000459965 |
| 58 | 0.000467235 |
| 60 | 0.000473772 |
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
Try the question before revealing its answer. These are reading checks, not trading certification or a site completion record.
What is the displayed Vanna in this example?
Reveal the worked answer
0.000022799535 Delta change per volatility point.
Always state the unit.
Does model agreement prove the same live-market change?
Reveal the worked answer
No.
Actual inputs, instrument features and executable quotes can differ.
Translate Vanna into its specific input, affected quantity and unit before applying it. Revalue when the local approximation no longer fits.
Try it in the working lab
- Record Delta, then raise IV from 25% to 26%.
- Compare that Delta change with Vanna per point.
- Return to the starting state; inspect the Vanna cross-term only when both spot and IV shocks are nonzero.
Your live inputs are independent of the fixed worked example above. Return to the “Lesson example” preset to reproduce it.
My reading record
Personal review marks are not quiz passes, course certification or trading approval. Old plugin progress is not overwritten.
Sources, assumptions and downloadable script
External references support definitions and conventions. Worked scenarios and prose are original. No current quotes or dealer inventory are claimed.
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