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I am reviewing the steps of proof that the binomial model is complete and don't understand the marked in red transition. Could anybody explain this step?

If $P^{**}$ is a risk-neutral measure, so that

$E^{**}[\bar{S}_{n+1} | \ F_n] = \bar{S}_n \ \ $ for all n

So given the structure of the model,

$\frac{1}{(1 + r)^{n+1}}E^{**}[S_{n+1} | \ F_n]=\frac{1}{(1 + r)^{n+1}}(uS_nP^{**}[R_{n+1}=u \ | \ F_n]+dS_nP^{**}[R_{n+1}=d \ | \ F_n])=$

$=\color{red}{\bar{S}_{n} \{ d+(u-d)\ P^{**}[R_{n+1}=u \ | \ F_n] \} }$


by martingale condition
$\frac{1}{1+r} \{ d+(u-d)\ P^{**}[R_{n+1}=u \ | \ F_n] \}=1$

so $P^{**}[R_{n+1}=u \ | \ F_n]=\frac{1+r-d}{u-d}=p^*$

$P^{**}=P^{*}$

Therefore the binomial model is complete.

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  • $\begingroup$ Please explain your notation. What is $P*$? $\bar{S}$ is the discounted stock price? Where is a reference to the above notation? In the red part, where does the curly bracket end? $\endgroup$ – Richard Jan 19 '16 at 11:40
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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.

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