Question

A European call option and put option on a stock both have a strike price of $21 and an expiration date in 4 months. The call sells for $2 and the put sells for $1.5. The risk-free rate is 10% per annum for all maturities, and the current stock price is $20. The next dividend is expected in 6 months with the value of $1 per share.

(a) describe the meaning of “put-call parity”. [2 marks]

(b) Check whether the put-call parity holds. [2 marks]

(c) If the put-call parity does not hold, describe step-by-step how an investor can take the advantage of this arbitrage opportunity to make a profit. [7 marks]

(d) If these two options are American options rather than European options, briefly explain whether the put-call parity still holds here. [2 marks]

Answer 1

a) Put Call parity is the relationship between the call and put prices on the same underlying asset with the same strike price and maturity. This relationship must hold to avoid arbitrage.

The relationship is

Call option price + present value of strike price = Put option price+Stock price (less present value of dividends within maturity of option)

b) In this case

Call option price + present value of strike price =  2+ 21/(1+0.1*4/12) = 22.32258

Put option price+Stock price (less present value of dividends within maturity of option)

= 1.5+20 (there is no dividend within maturity)

=21.5

As the two values are not equal, Put-Call parity does not hold

c) Arbitrage steps are as given below :

i) Borrow $19.5 for 4 months. maturity amount = 19.5*(1+0.1*4/12) = $20.15

ii) Sell call option at $2

iii) With $19.5+$2 = $21.5, buy the stock and the put option for $20 and $1.5 respectively

iv) After 4 months, if Stock price < $21, exercise the put option and sell the stock for $21. Pay off the loan amount of $20.15 and get the remaining amount of $0.85 as arbitrage profiit

v) After 4 months, if Stock price > $21, call option will be exercised .sell the stock for $21. Pay off the loan amount of $20.15 and get the remaining amount of $0.85 as arbitrage profiit

Thus in all situations, an arbitrage profit of $0.85 can be made after 4 months

d) If the options are American options, put call parity does not apply. This is because the American options can be exercised at any time before the expiry. Hence the present value of variables depend on the time of exercise which is not known. Thus , put call parity does not hold for American options