= Stock-numeraire measure in a binomial market
Let $Q$ be the risk-neutral measure in an $N$-period binomial market. The stock-numeraire measure is defined on terminal events by
$$
\widehat Q(A)=\frac{\mathbb E_Q[S_N\mathbf1_A]}{S_0(1+r)^N}.
$$
If the $Q$-probability of an up move is $q$, its up probability under $\widehat Q$ is
$$
\widehat q=\frac{q(1+b)}{1+r}.
$$
This change of measure converts discounted expectations containing a factor $S_N$ into probabilities under $\widehat Q$.
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