# How to calculate Vomma of Black Scholes model

This source (PDF) gives the closed-form for vomma (or volga, i.e. the second derivative of price w.r.t. volatility) of the Black Scholes option pricing model as:

$$S_{0}e^{-qT}\sqrt{T}\frac{1}{\sqrt{2\pi}}e^{-\frac{d_{1}^{2}}{2}}\frac{d_{1}d_{2}}{\sigma}$$

where

$$d_{1} = \frac{ln(S_{0}/K)+(r-q)T + \sigma^{2}/2T}{\sigma\sqrt{T}}$$

and

$$d_{2} = \frac{ln(S_{0}/K)+(r-q)T - \sigma^{2}/2T}{\sigma\sqrt{T}}$$

Two questions:

• Is this correct? Please provide additional source and/or proof.
• What is $q$? (it's not defined in the referenced document)

Edit: I think there's a missing set of parentheses around $\sigma^{2}/2$ in the formulas for $d_{1}$ and $d_{2}$. E.g. $d_{1}$ should be

$$d_{1} = \frac{ln(S_{0}/K)+(r-q)T + (\sigma^{2}/2)T}{\sigma\sqrt{T}}$$

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q is the yield. –  b.gatessucks Jan 18 at 19:50
oh, right, dividend yield, correct? –  laslowh Jan 18 at 19:57
Yes, it is right. Notice that it is much better to write (T - t) instead of T. –  edouard Jan 22 at 0:03
Your updated $d_1$ looks reasonable. I wonder then if $q=\sigma^2$. –  chrisaycock Jan 22 at 16:12
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## 2 Answers

That looks about right

Volga: S*Sqrt(T)*d1*d2*N'(d1)/σ

Edit: I provided a link to a pdf of the following book:

http://books.google.co.jp/books/about/The_complete_guide_to_option_pricing_for.html?id=tuoJAQAAMAAJ&redir_esc=y

but took it off because it was a scanned version and I was not sure it infringes on copyrights.

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Please restrain from posting links to copyrighted material. –  Alexey Kalmykov Jan 19 at 19:51
and you would be the expert on knowing this has not been put up with the explicit permission of the author or publisher? It seems to be part of the Boston U. url extension but I admit I have not asked anyone explicitly for permission to post a link to a link to a pdf file. Maybe what Google does is also illegal? Before you accuse me of horrible crimes please provide some sort of evidence of wrongdoing. Simply because a book appears in electronic form in the public internet domain in 2013 really means nothing. –  Matt Wolf Jan 20 at 8:56
please do not get me wrong, I am happy to immediately take down the link should someone show evidence that this is breaching any sort of copyright laws. So far I have not seen a thing other than your accusation. –  Matt Wolf Jan 20 at 8:57
This is not an accusation and you take it too personally for some reason. Indeed, many authors and publishers allow to download their books online, e.g. Elements of Statistical Learning. However, note that 1) this ebook is scanned and not the original PDF 2) author's site doesn't indicate existence of free electronic version of this book. I leave at your discretion as I have no interest in going further into further debate on this. –  Alexey Kalmykov Jan 20 at 14:27
@AlexeyKalmykov, well fair points made, especially the scan. Thanks for pointing it out. –  Matt Wolf Jan 20 at 14:32
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Vomma, or Volga or DvegaDvol is the second derivative of the option w.r.t volatility. In other words, it is the sensitivity of vega to changes in implied volatility.

A simple way to remember how Vomma is computed in the Black-Scholes framework is as follows: $$\frac{\partial^2 C}{\partial \sigma^2} = Vega \left(\frac{d_1d_2}{\sigma}\right)$$

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