This should be a standard exercise involving high-school Calculus, but for some reason, my expression for the European option volga, does not match the one on Wikipedia. I would like to ask if, someone can check my work.
Let $v(S_0,K,r_{FOR},r_{DOM},t,T,\sigma)$ be the value of the $T$-maturity option at time $t$.
The analytic vega of an option is:
$$ \frac{dv}{d\sigma} = S_0 e^{-r_{FOR}\tau} \phi(d_{+}) \sqrt{\tau} \tag{1} $$
Differentiating with respect to $\sigma$, we get:
$$ \begin{align*} \frac{d^2 v}{d\sigma^2} &= \frac{\partial}{\partial \sigma} \left(S_0 e^{-r_{FOR}\tau} \phi(\omega d_{+}) \sqrt{\tau}\right)\\ &=S_0 e^{-r_{FOR}\tau} \sqrt{\tau} \cdot \frac{\partial}{\partial \sigma} \phi(\omega d_{+})\\ &=S_0 e^{-r_{FOR}\tau} \sqrt{\tau} \cdot \frac{\partial}{\partial d_{+}}(\phi(\omega d_{+}))\cdot \frac{\partial d_{+}}{\partial \sigma} \\ &=S_0 e^{-r_{FOR}\tau} \sqrt{\tau} \cdot (-d_{+})\phi(\omega d_{+})\cdot \frac{\partial}{\partial \sigma} \left(\frac{\log F/K}{\sigma \sqrt{\tau}} + \frac{1}{2}\sigma \sqrt{\tau}\right)\\ &=S_0 e^{-r_{FOR}\tau} \sqrt{\tau} \cdot (-d_{+})\phi(\omega d_{+})\cdot \left(-\frac{1}{2}\frac{\log F/K}{\sigma^2 \sqrt{\tau}} + \frac{1}{2}\sqrt{\tau}\right)\\ \end{align*} $$
But this, doesn't seem to yield an expression involving $d_{+}$ and $d_{-}$, even when I tried doing so.