# Determining Presence of Arbitrage

I am slightly confused by part (b) of this question. My understanding is that the easiest way to determine if there is arbitrage is to compute the state prices and then look at their sign: if one or more of the state prices are nonpositive, then we have arbitrage.

We have a payoff matrix with 3 linearly independent securities, the stock, the bond, and either the first or second call option. What puzzles me is that we have prices at $$t=0$$ for the stock and bond but not for the call options. To my understanding, there are two ways to approach this question.

1. Just use the stock and the bond (incomplete market) to compute the state prices and obtain a range for the state prices. Fix the range such that none of the state prices are 0 or negative. Then argue that if the state prices fall outside of that range there will be arbitrage.
2. Use the stock, the bond, and an option and compute the state prices as a function of the unknown option price at $$t=0$$, $$C_0$$. Again, fix a range for $$C_0$$ such that none of the state prices are 0 or negative, and argue that whenever the option price $$C_0$$ lies outside of that range there will be arbitrage.

Does this sound reasonable? Does anyone have any other ideas as to how to answer this question?

• How about buying four units of the bank account and selling 1 unit of the stock? – Magic is in the chain Oct 18 '20 at 18:50
• That makes a lot of sense and is much simpler than what I was planning on doing. Just out of curiosity, is there a more rigorous way to show there is arbitrage, e.g. by doing something similar to what I had initially proposed? Thank you. – John Paris Oct 18 '20 at 21:47