Vanna Formulas and Unit Conventions in Black-Scholes Greeks

Summary

The document asks why two stated formulas for option vanna appear to differ by a factor of one hundred. It gives expressions involving spot price, volatility, time to maturity, the Black-Scholes variables, and vega, then reports that substituting a particular set of inputs produces values separated by that factor. The issue is presented as a discrepancy in the formulas or their interpretation.

The question highlights the importance of checking how quantities such as volatility, time, and vega are scaled when comparing Greek formulas. For example, a volatility expressed as a percentage versus a decimal, or vega quoted per volatility point versus per unit volatility, can change numerical values. However, the document provides no answer, derivation, or independent verification of the calculations, so it does not establish which convention explains the discrepancy.

Key ideas

• Vanna relates changes in option delta to changes in volatility and has equivalent formula representations under consistent definitions.
• A factor-of-one-hundred mismatch can arise when inputs or Greeks use different scaling conventions.
• The example reports a discrepancy but does not identify its cause.
• The formulas should be compared using consistent volatility, time, and vega units.

Tags

# The Wikipedia formulas for Vanna differ by a factor of 100x, why is that?


# The Wikipedia formulas for Vanna differ by a factor of 100x, why is that?












For a given:

- Stock price ${\displaystyle S\,}$ Strike price ${\displaystyle K\,}$ Risk-free rate ${\displaystyle r\,}$ Annual dividend yield ${\displaystyle q\,}$ Time to maturity ${\displaystyle \tau =T-t\,}$ (represented as a unit-less fraction of one year), Volatility ${\displaystyle \sigma \,}$ Vega $\mathcal {V}$

Vanna can (in my understanding) be computed from either of the 2 formula below:

rewritten in latex as

$$-e^{-qt}*\phi(d_1)*d_2/\sigma =\mathcal {V}/S*[1-(d_1/ (\sigma*sqrt(t))]$$

which solves into: $$ABC = ABC/100$$

or with the inputs given below, it solves into: $$ .03421= .0003421$$

which seems like it should be wrong.

Inputs I'm using to test are

- Stock price ${\displaystyle S\,}$ = $16.31 Strike price ${\displaystyle K\,}$ = $15.00 Volatility ${\displaystyle \sigma \,}$ = 123.58% Time to maturity ${\displaystyle \tau =T-t\,}$ = 19.319% of a year or 70.5151 days Risk-free rate ${\displaystyle r\,}$ = .33% Annual dividend yield ${\displaystyle q\,}$ = 0% Vega $\mathcal {V}$ = .026109 (based on my calculations)

Any ideas on what the difference is?

God bless,

NN

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.