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Let us have the standard single-period binomial pricing model, and denote the up and down states of the underlying by $S_u$,$S_d$ respectively. Let us say we have a call option on the underlying with strike $K$ such that: $$ K < S_d < S_u $$ I.e., an option that will surely be in the money. This (apparently, according to these notes: http://galton.uchicago.edu/~lalley/Courses/390/Lecture1.pdf) violates the no-arbitrage assumptions of the model.

Question: What is the arbitrage opportunity here? It seems that if the option is expensive enough there will be no arbitrage.

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  • $\begingroup$ Arbitrage will indeed depend on the price vs payoff, introducing this security might introduce redundancy but not necessarily an arbitrage if the price is right. Is K meant to represent the price of the risk free bond? $\endgroup$ Oct 14 '20 at 19:39
  • $\begingroup$ $K$ here is the price at which the option is struck at, so we have a world where the option is in the money in all states. $\endgroup$ Oct 14 '20 at 19:40
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    $\begingroup$ Ok. It is an uninteresting case, but agree that it is not necessarily an arbitrage unless the option is cheap! $\endgroup$ Oct 14 '20 at 19:43
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    $\begingroup$ Got it - I've attached the set of notes where I read that this would lead to an arbitrage. $\endgroup$ Oct 14 '20 at 19:49
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    $\begingroup$ I see - prior to that there's the statement that no one would sell the call if $K < d1$ or buy it if $K > d2$. $\endgroup$ Oct 14 '20 at 20:12

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