Question

Question 16 A stock currently trades at $52. It is expected that dividends of $1.00/share will be paid to owners of the stock at 1 month and at 4 months from the current date. Consider these dates as ex-dividend dates as well. The continuously compounded risk free rate is 5%. European call and put options on the stock with exercise prices of $50 and 6 months to the expiration date are currently trading. a) Calculate the lower bound for the value of the European call. (1 mark) b) How would you arbitrage if the European call option has a market price (premium) of $1.00? In your answer clearly identify your position in each relevant instrument. (1 mark) c) If the European call option has a market price (premium) of $2.00, based on put-call parity, what should be the price of a European put on the stock with the same exercise price and time to expiration? (1 mark) d) Calculate the lower bound for the value of an American call option on the stock with an exercise price of $50 and a time to expiration of 6 months. (1 mark)

Answer 1

a) Price of European call option c > max (0, S₀ - X *e^-rt)

X = strike price = 50, r= continuously compounded risk free rate = 5% =0.05, t= time in years

Where S₀ = adjusted Spot price = spot price - present value of dividends

= 52 - 1*e^{- 0.05* 1/12} - 1* e ^{- 0.05*4/12}

=52 - 0.9958- 0.9835

= 50.0207

So, Value of European Call (c) > max (0, 50.0207- 50*e^-0.05*6/12)

= max (0, 53.9793 -48.7655)

= 1.2552

Hence, the lowest bound is $1.26 for the European call option

b) If the call option is priced a $1 , then we can borrow a share and sell short, @ $ 52. Out of this, $1.9793 is the present value of dividend which you pay to borrower today, so, you are left with $50.0207, with this we buy a call option @$1 and invest the remaining $49.0207 at risk free rate.

After 6 months, we have $49.0207 * e^0.05*6/12 = $50.2617

If stock price >$50, we use the call option and buy the stock @ $50 and return it thereby saving $0.2617 as arbitrage profit

If stock price < 50 , we buy the stock from market at a price <50 and enjoy even more arbitrage profit

c) Put call parity equation implies

p+S₀ = c + Xe^-rt

=> p = 2+ 50*e^-0.05*6/12 - 50.0207

=2+48.7655 - 50.0207

= $ 0.7448