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I have the following question:

Why would somebody be interested in the expression $E[S^\theta]$ for $\theta$ between zero and one. The only thing I know is that this then can be somehow used to compute the call option price $(S-k)^+$.

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    $\begingroup$ I suspect you may be struggling with the concept of 'characteristic function', which is defined as \begin{align} \psi_{\log S_T}(u) &= E_t \left[ \exp\left\{ iu \log \left(\frac{S_T}{ S_t}\right)\right\} \right] \\ &= E_t \left[ \left(\frac{S_T}{S_t}\right)^{iu} \,\right] \end{align} with $u\in \mathbb R$. Fourier inversion of the characteristic function will give you the distribution, which you can then use to calculate the option price. $\endgroup$
    – user34971
    Oct 29, 2021 at 11:49
  • $\begingroup$ Thank you Frido, I do understand that concept. The issue I am having is that $\theta$ is only defined for the interval $(0,1)$ so I do not know what $S_T^{iu}$ is. $\endgroup$
    – Oli Bernet
    Oct 29, 2021 at 13:03
  • $\begingroup$ Ok, well I have not come across this particular problem before. As mentioned before, if you can give more details then perhaps I or others can be more helpful. $\endgroup$
    – user34971
    Oct 29, 2021 at 13:07

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