Question 20.ss.2: The stock of Augusta Light and Power is currently selling at...
The stock of Augusta Light and Power is currently selling at $12 per share. Over the next year the company is undertaking a new electricity production project. If the project is successful, the company’s stock is expected to rise to $24 per share. If the project fails, the stock is expected to fall to $8 per share. The risk free rate is 6 percent. Calculate the value today of a one year call option on one share of Augusta Light and Power with an exercise price of $20.
First determine the payoffs for the stock, a risk free loan, and the call option under the two possible outcomes. In one year, the stock price is expected to be either $8 or $24. The loan will be worth $1.06 regardless of whether the project is successful. If the project fails, the stock price will be less than the exercise price of the call option. The option will not be exercised, and will be worth $0. If the project is successful, the stock price will be higher than the exercise price of the call option. The option will be exercised and its value will be the difference between the stock price and the exercise price, $4.
The stock and loan can be used to create a replicating portfolio which has the same payoff as the call option:
($8 × x) + (1.06 × y) = $0
($24 × x) + (11.06 × y) = $4
Solving the two equations yields: x = 0.25, y = −1.887
The value of the call option is the same as the current value of this portfolio:
($12 × 0.25) + ($1 × −1.887) = $1.11