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If the condition $$0<d<1+r<u$$ is not satisfied, the Binomial model (with $d<u$) would have immediate arbitrage opportunity: 1) $1+r\geq u$: Then the riskfree asset would yield least as much return as the stock in any state for any probability, so you could short the stock to buy the riskfree asset and end up with some riskless profit with ...


This sounds like the first chapter of Björks book am I right? It treats a single-stage model. Simply put, if $1+r \leq d$ you buy the stock and have $V_1\geq 0$ with positive probability of making a profit. If $1+r \geq u$ you want to sell the stock short and buy the bond from the proceeds. The result is the same. Edit: To show that the condition is ...

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