Suppose $S_t$ is a geometric brownian motion. Then how to understand its time integral, i.e., $Y_t=\int_0^{t}S_udu$?

Is $Y_t$ still a stochastic process?

How to compute the expectation of $Y_t$?

How to compute the expectation of $\left[a-Y_t\right]^+$?

Here $[x]^+=\max\{x,0\}$.

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    $\begingroup$ Search for "Asian option" in the site's search bar, there is plenty of Q&A addressing these issues. $\endgroup$ – Daneel Olivaw Oct 30 '17 at 14:39
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    $\begingroup$ Daneel Olivaw is right. That being said, yes $Y_t$ is a stochastic process. The expectation operator being linear computing the expectation should be straightforward (just swap integral and expectation operators). For the non-linear payout you won't have a closed form result because the distribution of $Y_t$ is not analytically tractable, there are other options though (PDE, Monte Carlo). $\endgroup$ – Quantuple Oct 30 '17 at 16:07
  • $\begingroup$ The two are right. If you look at my answer to your original question (I suppose the two are connected) then you I really think you need the Asian option for Levy processes. $\endgroup$ – Richard Oct 30 '17 at 18:17
  • $\begingroup$ Thnks. Could you elabrate more why Asian option for Levy processes? $\endgroup$ – Zhiyuan Wang Oct 31 '17 at 0:55

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