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Note that \begin{align*} E\bigg(\frac{S_{i+1}}{S_i}\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) &=zE\bigg(\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg)-E\bigg(\Big(z-\frac{S_{i+1}}{S_i}\Big)\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) \\ &=zP\bigg(\frac{S_{i+1}}{S_i}<z\bigg)-E\bigg(\Big(z-\frac{S_{i+1}}{S_i}\Big)^+\bigg). \end{align*} Then you ...


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This is a very broad question and a large number of issues have been discussed in the literature. As such, it's hard to give specific advice except that it is better to model returns instead of prices directly. What I would do if I were you: Read some of the available literature to get a good overview. This is an interesting paper but many more exist. ...


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What you are looking for is the partial expectation of $\frac{S_{i+1}}{S_i}$. Since $\frac{S_{i+1}}{S_i}$ is lognormally distributed, you can use the following result: For a lognormal random variable $X \sim LND(m,v^2)$, $$ E(X | X < z) = E[X] \Phi\left( \frac{\log(z)-m-v^2}{v} \right) $$ In your case, $m = (r-\frac{1}{2}\sigma^2) (t_{i+1}-t_{i})$, $v^2 ...


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Number one, the central limit theorem means a lot of things that may not be normal end up looking normal when lots of little 'experiments' or impacts are added up. Number 2, when dealing with finance you need a model that seems plausible. An arithmetic Brownian motion could go negative, but stock prices can't. On the other hand, it seems quite plausible ...



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