Statistical distribution of a product of lognormal variables Question
I have two correlated lognormal variables $S_1$ and $S_2$, such that
$$\begin{pmatrix}
\ln S_1 \\ \ln S_2
\end{pmatrix}
\sim \mathcal{N}\left( \begin{pmatrix}
\mu_1 \\ \mu_2
\end{pmatrix}, \begin{pmatrix}
\sigma_1^2 & \rho \sigma_1 \sigma_2\\ \rho \sigma_1 \sigma_2 & \sigma_2^2
\end{pmatrix}\right)
$$
I would like to find the statistical distribution of the product $(S_1)^\alpha \, \times \, (S_2)^\beta$ where $\alpha, \beta \in \mathbb{R}$. I also need to find its expectation and standard deviation.
Workings
So far, I found that the distribution is going to be lognormal, with mean = $\alpha \mu_1 + \beta \mu_2$. However, I can't seem to figure out how the standard deviation is going to change... Any help on this? 
 A: As you nearly correctly mentioned,
$$
\mathbb{E} (\alpha \ln S_1 + \beta \ln S_2) = \alpha \mathbb{E}(\ln S_1) + \beta \mathbb{E} (\ln S_2) = \alpha \mu_1 + \beta \mu_2
$$
Note that 
$$
\begin{aligned}
\mathrm{Var}(\alpha \ln S_1 + \beta \ln S_2)&= \alpha^2 \mathrm{Var}(\ln S_1) + \beta^2 \mathrm{Var}(\ln S_2) + 2 \alpha\beta \mathrm{Cov} (\ln S_1, \ln S_2) \\
&= \alpha^2 \sigma_1^2 + \beta^2 \sigma_2^2 + 2 \alpha\beta \rho \sigma_1\sigma_2
\end{aligned}
$$
Thus we have
$$
\alpha \ln S_1 + \beta \ln S_2 \sim Normal(\alpha \mu_1 + \beta \mu_2, \alpha^2 \sigma_1^2 + \beta^2 \sigma_2^2 + 2 \rho\alpha\beta \sigma_1\sigma_2)
$$
and we have
$$
S_1^\alpha S_2^\beta = \exp (\alpha \ln S_1 + \beta \ln S_2) \sim LogNormal(\alpha \mu_1 + \beta \mu_2, \alpha^2 \sigma_1^2 + \beta^2 \sigma_2^2 + 2 \alpha\beta \rho \sigma_1\sigma_2)
$$
From the properties of LogNormal, 
$$
\mathbb{E} (S_1^\alpha S_2^\beta) = \exp(\alpha \mu_1 + \beta \mu_2 + \frac{\alpha^2 \sigma_1^2 + \beta^2 \sigma_2^2 + 2 \alpha\beta \rho \sigma_1\sigma_2}{2}) \\
\mathrm{Var} (S_1^\alpha S_2^\beta) = \left[ \exp (\alpha^2 \sigma_1^2 + \beta^2 \sigma_2^2 + 2 \rho\alpha\beta \sigma_1\sigma_2) - 1 \right] \times \\ \exp(2\alpha \mu_1 + 2\beta \mu_2 + \alpha^2 \sigma_1^2 + \beta^2 \sigma_2^2 + 2 \rho \alpha\beta \sigma_1\sigma_2)
$$
