Given $Z(t) \sim \mathcal{N}(0,\,1)$, why does the formula for the expected value of a lognormal random variable, $E(e^{X}) = e^{\mu + \sigma^2/2}$ give the following:
$$E(e^{\sigma \sqrt{t} Z}) = e^{\sigma ^2 t/2}$$
Since $\sigma = 1$, I thought the calculation would be $E(e^{\sigma \sqrt{t} Z}) = e^{(\sigma \sqrt{t}) \frac{1}{2} 1^2}$. Can anyone explain how to use the formula correctly?
Best Answer
HINT
Your formula states that if $X \sim \mathcal{N}\left(\mu, \sigma^2\right)$, then $$ \mathbb{E}\left[e^X \right] = e^{\mu + \sigma^2/2}. $$
In your specific example, if $Z \sim \mathcal{N}(0,1)$, then $\sigma \sqrt{t} Z \sim \mathcal{N}\left(0, \sigma^2t\right)$.
Can you now plug into the formula above?