Bounds on a call spread, spot the arbitrage

Question

Two call options on the same stock, same expiry: the $100$ -strike call is quoted at $\$ 7 $and the$ 110 $-strike call at$ $9$.

Is there an arbitrage? If so, construct it. What are the model-free bounds a call spread must obey?

Accepted Answer

Monotonicity in strike. A call with a lower strike gives you the right to buy at a lower price, so at every possible expiry price ST its payoff is at least as large as a higher-strike call's: \max(ST - 100, 0) \ge \max(ST - 110, 0) \quad 	ext{for all } ST. Therefore the 100-call must be worth at least as much as the 110-call. Here the market has it backwards: \7 < \9. That is a clean arbitrage. Construct the trade. Buy the cheap, sell the rich: - Buy the 100-call for \7, sell the 110-call for \9. Net credit of \2 today. At expiry the spread's payoff is \max(ST - 100, 0) - \max(ST - 110, 0), …

Bounds on a call spread, spot the arbitrage

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