A one-year European call on a non-dividend stock at 100 with strike 80 is quoted at 22.00 when the continuously compounded rate is 5%. Is that possible, and if not what do you trade and what is the profit?

A one-year European call on a non-dividend stock at 100 with strike 80 is quoted at 22.00 when the continuously compounded rate is 5%. Is that possible, and if not what do you trade and what is the profit?

Approach: Compare the quote to the model-free lower bound on a European call, which is the spot less the discounted strike, and construct the trade that captures the gap.

1.90. The model-free lower bound on a European call is S - K*e^{-rT} = 100 - 80*e^{-0.05} = 100 - 76.098 = 23.902, using the discount factor e^{-0.05} = 0.9512, and the quote of 22.00 sits 1.902 below it, so the price is an arbitrage unless there is a borrow cost or a dividend. Buy the call for 22.00, short the stock at 100 and lend 76.098 at 5%: the loan returns exactly 80 at expiry, which pays the strike if the call finishes in the money and the short is closed with the delivered share. If the stock finishes below 80 the short is closed in the market for less than 80 and you keep the difference, so the payoff is never negative and 1.902 was collected today. This bound is stronger than the intrinsic value bound of 20.00 because holding a call defers payment of the strike, and it is exactly the bound that fails once a hard borrow makes shorting the stock impossible.

Follow-up: What does the same bound become for an American call, and why does that make an American call on a non-dividend payer never worth exercising early?

Key concepts: lower bound, arbitrage, discount factor, intrinsic value.