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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.


You ask what to do "if risk-neutral pricing does not hold". By this I assume you mean that the price of an option is not equal to its expected value under the risk neutral probabilities (these are the probabilities that are calculated by enforcing the condition tha the expected return on the stock is the risk free rate, which can only exist if d <1+r< ...

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