Question

You are interested in a six-month European put option with a strike price of $60 on a non-dividend-paying stock when the stock price is $55. Your calculations revealed that lower bound for this option is $3.1 and option is trading at $2.5. What position do you need in option and underlying stock for an arbitrage profit?

Answer 1

Lower bound of the put option price = PV(K) - S0 = Ke-rf x t - S0 = 60e-rf x 6/12 - 55

This lower bound is given as $ 3.10

Hence, 60e-rf x 6/12 - 55 = 3.10

Or, e-rf x 0.5 = 58.10/60 = 0.9683

Hence, rf = -ln(0.9683) / 0.5 = 6.44%

Hence,  position you need in option and underlying stock for an arbitrage profit:

• Borrow an amount equal to cost of the put option + stock price = 2.5 + 55 = 57.5 for 6 months
• Buy the option
• Buy the stock

Initial cash flows = 0

And the arbitrage profit, P = max (K - S, 0) + S - Borrowed amount x erf x t

= max (60 - S, 0) + S - 57.5e6.44% x 0.5

= max (60 - S, 0) + S - 59.38

Case 1: S < 60; hence 60 - S > 0; hence the put option will be exercised

P = max (60 - S, 0) + S - 59.38 = 60 - S + S - 59.38 = $ 0.62

Case 2: S > 60; hence, 60 - S < 0; hence put option will not be exercised

P = max (60 - S, 0) + S - 59.38 = 0 + S - 59.38 = S - 59.38 > 0.62 as S > 60

Hence, we make a riskless and certain profit at the end of 6 months without any initial investment. This is the arbitrage profit.