# Binomial Model Strike Price Assumption

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.

• 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? Oct 14 '20 at 19:39
• $K$ 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. Oct 14 '20 at 19:40
• Ok. It is an uninteresting case, but agree that it is not necessarily an arbitrage unless the option is cheap! Oct 14 '20 at 19:43
• Got it - I've attached the set of notes where I read that this would lead to an arbitrage. Oct 14 '20 at 19:49
• I see - prior to that there's the statement that no one would sell the call if $K < d1$ or buy it if $K > d2$. Oct 14 '20 at 20:12