# 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:

• 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

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

• q is the yield. – b.gatessucks Jan 18 '13 at 19:50
• oh, right, dividend yield, correct? – laslowh Jan 18 '13 at 19:57
• Yes, it is right. Notice that it is much better to write (T - t) instead of T. – tagoma Jan 22 '13 at 0:03
• Your updated $d_1$ looks reasonable. I wonder then if $q=\sigma^2$. – chrisaycock Jan 22 '13 at 16:12

Volga: S*Sqrt(T)*d1*d2*N'(d1)/σ

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

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