= Risk-neutral probability in a binomial market
{title2=$q=(R-d)/(u-d)$}
The <martingale> condition for the discounted <stock> is $quS_0+(1-q)dS_0=RS_0$. Therefore the up <risk-neutral probability> is
$$
q=\frac{R-d}{u-d},\qquad 1-q=\frac{u-R}{u-d}.
$$
With down factor $d=1+a$, up factor $u=1+b$ and $R=1+r$, this is equivalently $(r-a)/(b-a)$. For $d=0.8$, $u=1.2$, $R=1.05$, it gives $q=0.25/0.4=0.625$. Under the <risk-neutral measure>, the terminal <stock> price has <expected value> $(5/8)120+(3/8)80=105$, whose discounted value is 100. The physical <probability> $p$ need only satisfy $0<p<1$; it does not enter this calculation.
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