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I got stuck on this problem trying to solve $E[e^{\sigma\xi}]$ where $\xi$ is normal distributed with $\mu$ = 0 and variance = $\sqrt{T-t}$

I'm supposed to calculate the expected value mentioned above either by using Itôs lemma or Moment Generated Function. I've tried with the 2nd one but got stuck on an integral (just writing the integration part and not the coefficients): $\int_{-\infty}^{\infty} e^{-((z-s\sigma)^2)/2}$.

The solution says it should be $\sqrt{2\pi}$ but I have no idea how you get that...

Fourierstudent
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  • Bit confused what you are asking here, do you know what the MGF of a normally distributed random variable is, because once you do you can just read off the answer? But looking at the integral in your question you seem to be stuck deriving the normalising constant of the Gaussian pdf? – Nadiels Sep 02 '17 at 15:08
  • The exercise only consists of my first sentence in original post. Only adds two hints that either you use Itos lemma or MGF which I tried, but got stuck on the integral i mentioned earlier. Still no idea why it becomes sqrt(2*pi) but maybe there's a "rule" to use to get it? – Fourierstudent Sep 02 '17 at 15:48
  • That particular integral is really well covered, see for instance https://math.stackexchange.com/questions/9286/proving-int-0-infty-mathrme-x2-dx-dfrac-sqrt-pi2 – Nadiels Sep 02 '17 at 15:57
  • otherwise either having a look through for questions regarding the expectation of a log-normal distribution or deriving the MGF of a Gaussian rv should get you everything you wish to know about the expectation in the original question – Nadiels Sep 02 '17 at 16:00

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