Question

Suppose you are given the following prices: Current stock price is $42. One month maturity Call(X=40)=$4.5, Two month maturity Call(X=40)=$3. All options are American type. Is there a profit opportunity based on these prices? If so, what would you do? If not, why not?

Answer 1

The near month call option (1 month maturity) is quoting at a higher price than the far month (2 month) call option at the same strike price, this should not happen if the cost of carry of positive. Hence there is a possible arbitrage opportunity as below:

• We sell the 1 month call option and receive $4.5 and buy 2 month call option for $ 3. Net cash flow = $1.5
• At the end of 1 month:
  • If the stock price is below $40, the 1 month call option expires worthless and minimum is profit = $1.5
  • If the stock price is above $40 and let us denote this price as S₁, then the loss = (40-S₁); in this case the trader should short the stock at S₁ stock price and the cash flows will be S₁ - (40 - S₁) = 40
• At the end of 2 months, let us say the price is S₂
  • If the price S₂ > 40 but less than S₁, the profit on 2 month option = S₂ - 40 and cash outflow on stock = -S₂. The net cash flow (including the $40 received from shorting stock in month 1) will be S₂ - 40 +40 - S₂ = 0
  • If the price S₂>S₁ then the profit on 2 month option is = S₂ - 40 and cash outflow on stock = -S_2. The net cash flow (including the $40 received from shorting stock in month 1) will be S₂ - 40 +40 - S₂ = 0
  • If the S₂<40, then the 2 month option expires worthless. The cash flows on short stock will be - S₂ and since S₁ >40, there will be profit equal to S₁ - S₂
• Thus we see that in either case there will be the minimum profit of $1.5 (difference between the 1 month and 2 month call options price) and in no case will there be a loss.