Assuming that the price of the stock follows the model
$ S(t) = S(0) exp ( mt − (σ^2/ 2) t + σW(t) ) , $ where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some constants.
What is the expectation and variance of S(2t)?
Expectation:
$E[S(2t)]=E[S(0)exp(2mt-(t\sigma^2)+\sigma W(2t)] = $
$S(0)E[exp(2mt-(t\sigma^2)+\sigma W(2t))] = S(0)exp(2mt-\sigma^2 t)E[exp(\sigma W(2t)]$
using that $W(2t)$ is $N(0,2t)$ I get that $W(2t)=\sqrt{2t} Z$, where $Z$ is $N(0,1)$.
$S(0)exp(2mt-\sigma^2 t)E[exp(\sigma \sqrt{2t} Z)]$ =
$S(0)exp(2mt-\sigma^2 t)exp(\sigma \sqrt{2t})E[e^{Z}] =$
$S(0)exp(2mt-\sigma^2 t)exp(\sigma \sqrt{2t})$
Is this solution correct?
Variance: Assuming that the expectation is correctly solved I could just use that $Var(S(2t))= E[(S(2t))^2] - E[S(2t)]^2$ ?