 Discrete-time binomial market

At each node a positive stock price moves to $u{S}_t$ or $d{S}_t$, 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 $R\le d$, 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 $R\ge u$, 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.