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I am also interested in the answer to this question, and would like to expand a little bit on it as well. First of all, let me add some value in terms of a partial answer: There are restrictions on when short selling is allowed. According to the SEC, and the "Alternative Uptick Rule" short selling is not allowed on "a stock that has dropped more than 10 ...


What if you write $$ P[R_{n+1} = d|F_n] = 1 - P[R_{n+1} = u|F_n] ? $$ Let us write $P(u) = P[R_{n+1} = u|F_n]$ Then the part to show is $$ u \bar{S}_n P(u) + d \bar{S}_n (1-P(u)) $$ and this $$ \bar{S}_n \left(d +(u-d)P(u) \right), $$ where we just expanded terms and then extracted the coefficients.

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