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
$$d_{1} = \frac{ln(S_{0}/K)+(r-q)T + (\sigma^{2}/2)T}{\sigma\sqrt{T}}$$
q
is the yield. $\endgroup$ – b.gatessucks Jan 18 '13 at 19:50