# 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

• Be careful that definitions of Vega differ. For some Vega is the amount of money per underlying share that the option's value will gain or lose as volatility rises or falls by 1 percentage point i.e. by 0.01. For others it is 100 times larger, a change in vol of 100 percentage points i.e. a rise by by 1 in $\sigma$. The former definition is more common in practice. Which one are you using? Apr 5 at 19:10
• I think that's what it was. One of the websites that I used to help put together the spreadsheet, (Macroption), lists Vega's formula as including the divide by 100, which threw off the calculations for Vanna. Wikipedia seems to have the correct formula. Thank you for your help! Apr 5 at 23:17
• You are welcome. It is good that Wikipedia is internally consistent. Apr 6 at 7:53