Question

6. A stock that does not pay dividend is trading at $50. A European call option with strike price of $60 and maturing in one year is trading at $10. An American call option with strike price of $60 and maturing in one year is trading at $15. You can borrow or lend money at any time at risk-free rate of 5% per annum with continuous compounding. Devise an arbitrage strategy.

So I know that usually american calls are never exercised early b/c it is not optimal. however, for this problem we have to assume that the call will get exercised. I realize that part of the strategy could be to buy the European call for $10 and sell the American option for $15 to make a profit of $5. however, say that the person who owns the American call option exercises it early. then you have to give them a stock and they will give you $60. how do I go about from there to create the arbitrage strategy and show the profit

Answer 1

You are on the right track. However you just need little guidance. Please see if the explanation below helps:

Create the arbitrage portfolio as below:

1. Short (Sell) American Call option for $ 15
2. Long (Buy) European call option for $ 10
3. Borrow $ 45 at risk free rate
4. Long (Buy) one stock as $ 50 (Current price)

Net cash flow required to create this portfolio = + 15 - 10 + 45 - 50 = 0

At any time prior to expiration i.e. t < 1 year

• If the Buyer of the American call option exercises, square off the position by selling the one stock you have in your portfolio for $ 60.
• Immediately, use this amount to square off the borrowed position. Liability of the borrowed position will be = $ 45e^r_ft = 45e^0.05t. For t < 1, 45e^0.05t. will always be < 60. Hence, 60 - 45e^0.05t.≥ 0
• Thus you will be left with Long position in European call option whose payoff on expiration ≥ 0

Thus overall payoff =  60 - 45e^0.05t.+ max (S - K, 0) ≥ 0

Thus you have made money without any initial investment. This is the arbitrage.

On expiration and if S > K = 60

• A long position in European Call option and a short position in American Call option will nullify each other.
• You will be left with 1 stock and borrowed amount to be repaid along with interest.
• Hence, payoff will be: S - 45e^0.05. = S - 45 x (1 + 5%) = S - 47.31 ≥ 0 as S > K = 60

Thus you are making money in any case without any initial investment. This is the arbitrage.