Volatility of Multiple Stocks

Question

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

Answer 1

Use Ito's lemma on the function $f(x,y) = xy$ and then extract out the diffusion term.

Answer 2

To obtain the vola for the log returns is easy and you don't need itos lemma, since

$$ \log S'(t) = \log S_1(t) + \log S_2(t),$$ therefore $$ var(S'(t)) = var(\log S_1(t)) + var(\log S_2(t)) + 2covar(\log S_1(t),\log S_2(t))\\ = \sigma_1t+\sigma_2t+2\rho \sigma_1\sigma_2.$$

However, to get the vola for the non-log stock price you indeed need to use the ito formula.