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Black–Scholes Vomma as Vega Scaled by d1d2 over Volatility

Article Quant Q&A · Author: laslowh
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Summary

The document presents the Black–Scholes formula for vomma, also called volga, and asks whether the expression and its inputs are correct. Vomma is the second derivative of an option price with respect to volatility, so it measures how vega changes as implied volatility changes. The replies provide a compact identity: vomma equals vega multiplied by d1 times d2 divided by volatility. This gives a practical way to calculate the sensitivity from the familiar Black–Scholes vega and d1/d2 terms.

The question also flags missing parentheses in the displayed d1 and d2 formulas and asks what q represents. The answers affirm the relationship and identify equivalent terminology, but do not explain q or work through a derivation. In the usual dividend-yield formulation, q is the continuous yield on the underlying; the equations must be typeset with the volatility-squared term multiplied by time in the numerator. The result applies within the Black–Scholes assumptions and does not address model risk or alternative pricing frameworks.

Key ideas

  • Vomma, volga, and DvegaDvol refer to the second derivative of option value with respect to volatility.
  • In Black–Scholes, vomma equals vega multiplied by d1d2 and divided by volatility.
  • The d1 and d2 expressions require clear parentheses around the volatility term multiplied by maturity.
  • The replies confirm the identity but do not provide a derivation or define q in the original exchange.

Tags

OptionsDerivatives pricingVolatility
Full text
# How to calculate Vomma of Black Scholes model


# 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

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

## Answer by Matt Wolf (score 1)

https://quant.stackexchange.com/a/7029

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.

## Answer by tagoma (score 1)

https://quant.stackexchange.com/a/7054

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: $$\frac{\partial^2 C}{\partial \sigma^2} = Vega \left(\frac{d_1d_2}{\sigma}\right) $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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