= Discrete-time binomial market
{title2=$S_{t+1}\in\{uS_t,dS_t\},\quad B_{t+1}=RB_t$}
= Binomial market
{synonym}
At each node a positive <stock> price moves to $uS_t$ or $dS_t$, with $0<d<u$, while a <bank account> grows by a fixed gross factor $R>0$. Assume both successors have positive physical <probability>, no dividends, a <frictionless market> with no trading costs, divisible holdings, unrestricted <short selling>, and borrowing and lending at the same fixed $R$. A <self-financing portfolio> receives no outside cash after its initial funding; subsequent rebalancing uses its own proceeds.
This <binomial market> excludes <arbitrage> exactly when $d<R<u$. If $R\leq d$, borrowing $S_0$ to buy one share gives terminal gains $S_0(u-R)$ or $S_0(d-R)$; these are nonnegative and one is positive. If $R\geq u$, <short selling> one share and lending its proceeds gives the analogous <arbitrage>. Strict inequalities allow a positive <risk-neutral probability> for each state. The two traded payoffs also span both successor states, giving <market completeness> relative to the information generated by this tree.
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