Question

Part a) You are given the following positions in options: long a call with exercise price X1, short a call with exercise price X2, long a put with exercise price X2 and short a put with exercise price X1. Assume that X2 > X1. Graph the payoffs from the trading strategy above, where all the options have the same expiration date. Comment on the position and the net payoff.

b) A stock currently trades at £50 per share. Suppose that a 3-month European put option on this stock with exercise price £50 sells at £4. The risk-free rate is 10% per annum.

i) What is the price of an at-the-money call option with exercise price £50? Assume that the stock does not pay any dividends in the next three months.

ii) Suppose the price of the call option were £5 in the market. Calculate the arbitrage profit that can be made.

iii) Now assume that in the next period (three months from now), the price of the stock can either increase to £55 or decrease to £45. Draw a one-step binomial tree and calculate the price of the put according to the binomial pricing method. The risk free rate is still 10% per annum.

Answer 1

a) Have a look at the following diagram for the payoffs:

b)i) Let's solve this using Put call parity. As per put call parity:

Call - Put = Spot - Present value of strike price

In the given problem,

Call = 4 + 50 - (50 * e^(-0.1*0.25)) = 5.234504

ii) Arbitrage profit 3 months hence = 5.234504 - 5 = 0.234504

iii) Let us first find the price of the call option as per the binomial tree and the use put call parity to find out the price of the put option