Suppose the the value of an stock is $S_t = S_{t-1}exp(\mu +\sigma Z_t) $ where $Z_t$ are standard normal variables. Find the distribution of ln($S_1/S_0$) under the Q measure given that dQ/dP is $exp(-\theta_1Z_1 -\theta_2Z_2 -1/2(-\theta_1^2 +-\theta_2^2$))
My working:
Since E(X) under Q measure = E(XdQ/dP) under P measure
$S_1/S_0$ = $exp(\mu +\sigma Z_1)$
E($S_1/S_0$) = E($e^{(\mu +\sigma Z_1)} e^(-\theta_1Z_1 -\theta_2Z_2 -1/2(-\theta_1^2 -\theta_2^2$)))
E($S_1/S_0$) = $e^{(\mu -\theta_1^2 -\theta_2^2)}E(e^{(\sigma-\theta_1)Z_1})E(e^{(-\theta_2Z_2)})$
How do u find the expectation of $E(e^{(-\theta_2Z_2)})$? Not very sure how to find that?