Problem 8-3 Black-Scholes Model Assume that you have been given the following in
ID: 2613830 • Letter: P
Question
Problem 8-3
Black-Scholes Model
Assume that you have been given the following information on Purcell Industries:
According to the Black-Scholes option pricing model, what is the option's value? Round your answer to the nearest cent.
Binomial Model
The current price of a stock is $16. In 6 months, the price will be either $20 or $11. The annual risk-free rate is 5%. Find the price of a call option on the stock that has an strike price of $14 and that expires in 6 months. (Hint: Use daily compounding.) Round your answer to the nearest cent. Assume a 365-day year. Do not round your intermediate calculations.
Current stock price = $16 Strike price of option = $11 Time to maturity of option = 4 months Risk-free rate = 8% Variance of stock return = 0.14 d1 = 1.965949 N(d1) = 0.975348 d2 = 1.749924 N(d2) = 0.959934Explanation / Answer
Solution
Black-Scholes Model
Calculation of option's value:
Vc = P[N(d1)] – Xe-rRFt [N(d2)]
= $16(0.975348) – $11e-0.08*0.33 (0.959934)
= 15.605568 – 11 * 0.973945433 * 0.959934
= 15.605568 – 10.71339977 * 0.959934
= 15.605568 – 10.28415669
= 5.32
Here,
P = $16
N(d1) = 0.975348
X = $11
r = 8% = 0.08
t = 4 months = 4/12 = 0.33
N(d2) = 0.959934
Binomial Model
The stock’s range of payoffs in 6 months is $20 - $11 = $9
At expiration, the option will be worth $20 - $14 = $6, if the stock price is $26, and zero if the stock price is $11.
The range of payoffs for the stock option is $6 – 0 = $6
Equalise the range to find the number of shares of stock:
Option range / Stock range = $6 / $9 = 0.666666666
With 0.666666666 shares, the stock’s payoff will be either $13.33333332 (20*0.666666666) or $7.333333333 (11*0.666666666).
The portfolio’s payoff will be $13.33333332 – 6 = $7.33333332, or $7.333333333 – 0 = $7.333333333
The present value of $7.333333333 at the daily compounded risk-free rate is:
PV = FV / (1 + (r/k)n)
= $7.333333333 / (1 + (0.05/365))182.5 = $7.152284936
Here, FV = $7.333333333
r = 5% = 0.05
k (compounded times a year) = 365 times
n = 6 months = 182.5 days
The option price is the current value of the stock in the portfolio minus the PV of the payoff:
V = (0.666666666 * $16) - $7.152284936 = $ 3.51