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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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I simply priced the options based on the volatility,strike price, issue date, expiration date (the greeks) using a binomial price engine and then calculated the returns based around this

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In binomial tree models, there is no such a thing as a path. The binomial tree represents information about the distribution of the zero-curve at a given time and preserve enough information between different times to let you compute conditional expectations. Generally, you can not price path-dependant instruments in a model based on trees—because there is ...

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