Madeline Manufacturing Inc.’s current stock price is $45 per share. Call options

Question

Madeline Manufacturing Inc.’s current stock price is $45 per share. Call options for this stock exist that permit the holder to purchase one share at an exercise price of $35. These options will expire at the end of 1 year, at which time Madeline’s stock will be selling at one of two prices $25 or $55. The risk-free rate is 3.5%. Using the binomial option pricing model, create a riskless hedged investment and answer the following questions:

After the payoffs have been equalized and the riskless hedged investment is created, what is the value of the portfolio in one year? Round your answer to two decimals if needed.

$

Explanation / Answer

In this case, the current stock price is $45, and at the end of the year, the stock will be either $25 or $55. We also have a call option with a strike price of $35. The payoff for this call option at the end of the year is

                                                                 $20 given a stock price of $55 (Max ($55 - $35, 0))

Call price today

                                                                 $0 given a stock price of $25 (Max ($25 - $35, 0))

Let H represent the number of units of the stock we should hold. The investment would be H x $45 - call option price. If the stock price were $55 at the end of the year, the value of the call options would be $20. The payoff on our total investment would be H x $55 - $20. If the stock price at the end of the year were $25, the value of the call option would be $0. The payoff from the total position on the stock and call option would be H x $25 - $0.

We find the appropriate risk-free hedge, H, by setting the payoff in the up state equal to the payoff in the down state,

H x $55 - $20 = H x $25 - $0

H = ($20 - $0)/ ($55 - $25)

H = 0.6667

Solving for H1 , we find that the payoffs would be exactly the same in each state, if H is equal to .6667. For H = .6667, the payoff in the up state is .6667 x $55 - 20, or $16.6667. In the down state .6667 x $25 - 0, or $16.6667. Thus, by buying .6667 shares of the stock and writing a call option, we have created a riskless hedge with a payoff of $16.6667.

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