# Answer to Question #6325 in Economics of Enterprise for Renee

Question #6325

The current price of a stock is $22, and at the end of one year its price will be either $27 or $17. The annual risk-free rate is 6.0%, based on daily compounding. A 1-year call option on the stock, with an exercise price of $22, is available. Based on the binominal model, what is the option's value?
a. $2.43
b. $2.70
c. $2.99
d. $3.29
e. $3.62

Expert's answer

c. $2.99

The stock’s range of payoffs in one year is $27 - $17 = $10. At expiration, the option will be worth $27 - $22 = $5

if the stock price is $27, and zero if the stock price $17. The range of payoffs

for the stock option is $5 – 0 = $5. Equalize the range to find the number of

shares of stock: Option range / Stock range = $5/$10 = 0.5. With 0.5 shares, the

stock’s payoff will be either $13.5 or $8.5. The portfolio’s payoff will be

$13.5 - $5 = $8.5, or $8.5 – 0 = $8.5. The present value of $8.5 at the daily

compounded risk-free rate is: PV = $8.5 / (1+ (0.06/365))

minus the PV of the payoff: V = 0.5($22) - $8.005 = $3.00.

The stock’s range of payoffs in one year is $27 - $17 = $10. At expiration, the option will be worth $27 - $22 = $5

if the stock price is $27, and zero if the stock price $17. The range of payoffs

for the stock option is $5 – 0 = $5. Equalize the range to find the number of

shares of stock: Option range / Stock range = $5/$10 = 0.5. With 0.5 shares, the

stock’s payoff will be either $13.5 or $8.5. The portfolio’s payoff will be

$13.5 - $5 = $8.5, or $8.5 – 0 = $8.5. The present value of $8.5 at the daily

compounded risk-free rate is: PV = $8.5 / (1+ (0.06/365))

^{365}= $8.005. The option price is the current value of the stock in the portfoliominus the PV of the payoff: V = 0.5($22) - $8.005 = $3.00.

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