According to BSM, Stock Price follows log-normal distribution s.t. $$S(t)=S(0)*\exp(\sigma\sqrt t Z-(\sigma^2t)/2)$$ where Z is standard normal variable Then volatility of this stock is $\sigma \sqrt t$.
Suppose I model a new stock $S'(t)=S_1(t)*S_2(t)$ where $S_1(t)$ and $S_2(t)$ will follow log-normal distribution as mentioned above with parameters $\sigma_1,Z1$ and $\sigma_2,Z2$ and $Z_1$ and $Z_2$ are correlated with a factor rho, then how do I calculate volatility of new stock denoted by price S'(t) ?
Currently, according to How to calculate the volatility matrix with multiple stocks
I calculated volatilty = sqrt(sigma1^2+sigma^2)*sqrt(t) but I am not sure if it's correct especially for correlated Zs