Question #12661

4. 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

a. $2.43

b. $2.70

c. $2.99

d. $3.29

e. $3.62

Expert's answer

Answer is 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 in case the stock price is $27, and zero if the stock price is $17.

The range of payoffs for the stock option is $5 – 0 = $5. We 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 portfolio minus the PV of the payoff: V = 0.5($22) - $8.005 = $3.00.

The range of payoffs for the stock option is $5 – 0 = $5. We 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))

The option price is the current value of the stock in the portfolio minus the PV of the payoff: V = 0.5($22) - $8.005 = $3.00.

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