Question

A stock currently has a market value of $10. The risk-free rate of return is 3%. In one year, the stock is expected to sell for either $14 or $9. a) What is the value of a twelve-month call option with a strike price of $12? (6 points) b) If the twelve-month call option is traded at $1.2, Recommend a riskless strategy by showing the positions in both shares and calls. (4 points).

Answer 1

a)

Call option payoff if the stock price ends up at $14 = Stock price at maturity- Strike price = $(14-12) = $2

Call option payoff if the stock price ends up at $9 = $0

Present value of twelve-month call option = Expected payoff at maturity discounted at PV

The probability of the stock selling at $14 or $9 is same ie. 0.5

Present value of twelve-month call option = (0.5*2 + 0.5*0)*e^(-0.03*1) = $0.9704

Value of twelve-month call option = $0.9704

b) If the call option is trading at $1.2 then an arbitrage opportunity exists.

1) Short sell the call option and earn $1.2 upfront

2) Buy the stock by borrowing $(10-1.2) = $8.8 at risk-free rate 3%

If the stock ends up at $14

Sell the long stock for $14 = +$14

Repay the borrowed money = -$(8.8*1.03) = -$9.064

Loss from the short sell call = $(12-14) = -$2

Total profit = $14-$9.064-$2 = $2.936

If the stock ends up at $9

Sell the long stock for $9 = +$9

Repay the borrowed money = -$(8.8*1.03) = -$9.064

Payoff short sell call = $0

Total loss = +$9-$9.064+$0 = -$0.064

Profit from arbitrage at PV = (0.5*$2.936 + 0.5*(-$0.064))*e^(-0.03*1) = $1.3936