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I'm doing a mathematical finance module in discrete time. One of the lemmas given is $0<d<1+r<u$ there is no arbitrage opportunity. Where $d$ is a multiplier for when the stock price goes down from time $0$ to time $1$ and $u$ is the multiplier of the stock price going up.

So my question is why is that and if possible can you give an example where an arbitrage could exist and what it would look like say when $1+r = u$

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    $\begingroup$ Short a share and invest the cash at the riskless rate. If it goes up, you break even, if it goes down you win. You can't lose. $\endgroup$
    – lulu
    Commented Apr 14, 2022 at 19:11

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Let $1+r \le d < u$ with initial endowment of $0$. Borrow amount $S_0$ at time $t=0$, and buy a share. At the end of the period, you have amount $S_1 \in \{S_0d, S_0u\}$, and $S_0u > S_0d \ge S_0(1+r)$, so you can pay back the amount borrowed with a nonnegative profit.

Similarly, if $d < u \le 1+r$, sell one share short for amount $S_0$, lend that amount at $1+r$, and at $t=1$, you have $S_0(1+r) \ge S_0u > S_0d$, allowing you to buy back a share at the end of the period and close out the short trade with a nonnegative profit.

In either case, you have a positive expected value assuming both the up and down states have nonzero probability.

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