Corporate Finance, Module 23: “Advanced Option Valuation”
Black-Scholes Practice Problems
(The attached PDF file has better formatting.)
{This posting contains more information than is needed for the corporate finance on-line course.}
Exercise 23.1: Black-Scholes Pricing
A stocks price variance rate, or σ2, is 25%. (Brealey and Myers sometimes express this as the annual variance of a company’s continuously compounded stock price.) The nominal risk-free rate payable quarterly is currently 8%. (8% per annum with quarterly compounding is 2% each quarter.) The company’s stock now trades at $100. Three-month European calls and puts are trading with a strike price of $108.
A. What are the values of the five input parameters to the Black-Scholes model?
B. What is the value of ln(S/PV(X)): the logarithm of the ratio of the current stock price to the present value of the exercise price?
C. What are the values of d1 and d2?
D. What are the values of N(d1), N(–d1), N(d2), and N(–d2)?
E. What is the value of the European call option?
F. What is the value of the European put option?
G. Verify that the put call parity relation holds.
Solution 23.1:
Part A: We determine the Black-Scholes parameters:
● σ2 = 25% per year, so σ = 50%.
● t = 0.25
● r = 8% payable quarterly or 2% per quarter.
● S = $100
● X = $108 and PV(X) = $108 / 1.02 = $105.88
Part B: ln(S/PV(X)) = ln($100/$105.88) = ln(0.944) = -0.057
Part C: The values of d1 and d2 are
d1 = (-0.057 + ½ × 0.25 × 0.25 ] / (0.5 × 0.5) = -0.104
d2 = –0.104 – 0.5 × 0.5 = -0.354
Part D: N(d1) = N(–0.104) = 0.459; N(–d1) = N(0.104) = 0.541
N(d2) = N(–0.354) = 0.362; N(–d2) = N(0.354) = 0.638
Part E: The value of the call option is $100 × 0.459 – $105.88 × 0.362 = $7.57
Part F: The value of the put option is –$100 × 0.541 + $105.88 × 0.638 = $13.45
Part G: $7.57 + $105.88 = $13.45 + $100 = $113.45
Exercise 23.2: Black-Scholes Pricing
● The standard deviation of the continuously compounded annual rate of return on the stock is 0.4.
● The stock price is now $100 and pays no dividends.
● The time to maturity of the option is 3 months (0.25 years).
● ln (current share price / present value of the exercise price) = –0.08, at the risk-free rate.
A. What is the present value of the exercise price? (Derive this value from ln(S / PV(X)) = –0.08.) This is the one Black-Scholes parameter that we are not explicitly told.
B. What are the values of d1 and d2?
C. What are the values of N(d1), N(–d1), N(d2), and N(–d2)?
D. What is the value of the European call option?
E. What is the value of the European put option?
F. Verify that the put call parity relation holds.
Solution 23.2:
Part A: We determine the values of the Black-Scholes parameters:
● S = $100
● t = 0.25
● σ = 0.4
● ln(S / PV(X)) = –0.08 ➾ PV(X) = S / e–0.08 = S × e0.08 = $108.33
Part B: The values of d1 and d2 are
d1 = (–0.08 + ½ × 0.16 × 0.25 ] / (0.4 × 0.5) = -0.300
d2 = –0.300 – 0.4 × 0.5 = -0.500
Part C: The values are
N(d1) = N(–0.300) = 0.382; N(–d1) = N(0.300) = 0.618
N(d2) = N(–0.500) = 0.309; N(–d2) = N(0.500) = 0.691
Part D: The value of the call option is $100 × 0.382 – $108.33 × 0.309 = $4.73
Part E: The value of the put option is –$100 × 0.618 + $108.33 × 0.691 = $13.06
Part F: $4.73 + $108.33 = $113.06 = $13.06 + $100