A bit of background. I know that the forward price of a stock (or its expected price) is given by $\mathbb{E}[S_T]=S_te^{(r-q)(T-t)}$. Here, $r$ and $q$ are not constant, but follow a curve. I was wondering whether the following is true: $\mathbb{Var}[S_T]=S_t^2e^{\sigma^2(T-t)}$, where $\sigma^2$ is the Black-Scholes volatility. I believe this to be true, but I cannot convince myself.
Could anyone help me out on this?
Edit. Thanks for the help guys. I was also wondering whether it was possible to determine this value. $\mathbb{Var}[e^{(r-q)(T-t)}]$. Just that value without the stock price?