Question

A European call option c with a strike price K of £50 is traded for £32. The current value of the underlying asset S0 is £31. The interest rate r is 10% pa, and the time-to-maturity T is equal to six months. The underlying asset pays no dividends.
a) Which of the arbitrage bounds does the option value violate?

b) How would you profit from the violation of the arbitrage bound? Show the payoffs of your arbitrage

strategy assuming that the value of the underlying asset is either equal to £25 or equal to £55 at maturity. c) Show the payoff to the arbitrage strategy in a graph (with the payoff at maturity on the y-axis and the stock value at maturity on the x-axis).
d) If the option were a put and not a call option, would its value still violate the equivalent arbitrage bound for put options?

Answer 1

(a) Value of the European Call cannot be more than the value of the underlying asset. Here, the value of the European call option c is 32 while value of underlying asset is 31 , hence arbitrage is possible.

(b) Arbitrage is possible by buying the underlying asset and selling the call option. At the point of initiation , the net inflow is 1.

At Maturity if value of underlying is 25 , then the call option becomes worthless because the strike price is 50 which is higher than the maturity price. Hence the Total Payoff would be Value of Stock + Inflow at Initiation = 25 + 1 = 26

At Maturity if value of underlying is 55 , the value of call option would be ( Value of underlying - Strike Price ) = 55 - 50 = 5. Hence the total payoff would be Value of Stock - Value of Call Option + Inflow at Initiation = 55 -5 + 1 = 51

(c)

The payoff in the graph states that the payoff is 1 more than the underlying price till the underlying price attains the exercise price. Once exercise price is attained , the payoff remains constant at execise price + 1 which is 50 + 1 = 51.

(d) For European Put Option , Maximum Value is the present value of the exercise price = 50 / ((1+0.1)^(6/12)) = 50/(1.1^0.5) = 50/1.0488 = 47.6735

Hence , the arbitrage bound is not broken if the option were a put option.