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The answer assumes a normal distribution. As you can see in this graph, in the normal distribution: 50% of the outcomes are in the right hand part of the distribution (i.e. higher than the mean) 34% of the outcomes are between the mean and 1 standard deviation The question wants you to determine the probability that your stock returns less than 11%. If ...

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Let's just deal with the aspect of probabilities. The answer is easier than you think. Consider the simplest one-step model. At the end, the stock will either be up (to $Su$) or down (to $Sd$). It will move up with a probability p or down with a probability (1-p). Here's how to calculate p: $p = \frac {e^{rT} - d}{u - d}$ For example, consider a stock ...

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First find minimum value of j that assures the option is in the money. This would be function of u,d,n,S,K,p,q Any particular value of j has a probability associated with it. You gave formula above. Then you need sum probabilities from j=min j needed to n. See wikipedia binomial distribution to find formula for sum. Sum is based on cdf of binomial ...

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