Consider a European call option on Apple stock that expires one year from today. Suppose that the option has a strike price of $110 and that the current price of Apple stock is $100. There are two possible prices for Apple stock one year from today: $130 and $80. Suppose the annual risk free rate of interest is 10%.
What is the hedge ratio of the option? That is, how many shares of Apple stock do you need to buy as part of a portfolio of stock and risk-free bonds to replicate the payoffs from the Apple call option?
How much risk-free borrowing is necessary as part of the “replicating portfolio” for the Apple call option?
What is value today of the call option? (Hint: How much does it cost today to set up the replicating portfolio?)
Suppose that the call option were an American option and not a European option. Given the value of the option you computed in part (3), how large of a dividend payment would Apple have to make over the next year (i.e. before option expiration) for you to prefer to exercise the option today rather than to hold it to expiration. (Hint: Recall that the holder of a call option does not receive the dividend payments on the underlying stock.)