Question

Problem 8-3
Black-Scholes Model

Assume that you have been given the following information on Purcell Industries:

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) = 0.959934

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.

Answer 1

Solution

Black-Scholes Model

Calculation of option's value:

V_c = P[N(d₁)] – Xe^-r_RF^t [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