Question 20.A.1: Valuing a Call Option PROBLEM: You are considering purchasi...
Valuing a Call Option
PROBLEM: You are considering purchasing a call option on the stock of Grote Agricultural Company. Grote stock currently trades for $35 per share, and you predict that its price will be either $25 or $50 in one year. The call option would enable you to buy a share of Grote stock in one year for $30. What is this option worth if the risk-free rate is 4 percent?
APPROACH: The value of the option can be determined by computing the cost of constructing a portfolio that replicates the payoffs from that option.
With an exercise price of $30, the option will be worth $20 if the stock price rises to $50 ($50 − $30 = $20) and will be worth $0 if the stock price declines to $25. Therefore, the replicating portfolio for this option can be determined from the following two equations:
\begin{matrix} \$20 &=& (\$50 \times x)+(1.04 \times y) \\ \$0 &=& (\$25 \times x)+(1.04 \times y) \end{matrix}Solving for x and y, we find that x = 0.80 and y = −\$19.23. Therefore, the replicating portfolio consists of 0.8 share of Grote stock and a $19.23 loan. Since a 0.8 share would cost $28 (0.8 × $35 = $28), and $19.23 of this amount would be covered by the loan, this replicating portfolio would cost $8.77 ($28.00 − $19.23 = $8.77) to construct. Therefore, the call option is worth $8.77.