The one-year 100-strike call is 6.00 and the 105-strike call is 0.50 on the same underlying, with rates at 5% continuously compounded. Is that pair possible, and what do you trade?

The one-year 100-strike call is 6.00 and the 105-strike call is 0.50 on the same underlying, with rates at 5% continuously compounded. Is that pair possible, and what do you trade?

Approach: Bound the price difference between two strikes by the discounted strike difference, since that is the largest the spread can ever pay, and compare it to the quoted difference.

0.744. The call spread can pay at most the strike difference of 5.00 at expiry, so today it cannot be worth more than 5*e^{-0.05}, or 4.756. The quoted spread costs 6.00 - 0.50 = 5.50, which breaks the slope bound, so you sell the 100 call, buy the 105 call and invest the 5.50: at expiry you owe at most 5.00, whose present value is 4.756, leaving an arbitrage of 0.744 today. Stated as a slope, the derivative of the call price in strike must lie between -e^{-rT} and 0, which is the same statement as the risk-neutral density being a probability, and the discounted strike difference is what that bound integrates to. This check and the butterfly convexity check together are the two that keep a fitted smile arbitrage free.

Follow-up: What is the matching pair of bounds on put prices across strikes, and which one does a hard borrow break first?

Key concepts: call spread, slope bound, arbitrage, discounted strike difference.