Corporate Finance, Module 23: "Advanced Option Valuation"
Black-Scholes Practice Problems
(The attached PDF file has better formatting.)
Updated: July 25, 2007
{This posting contains more information than is needed for the corporate finance on-line course. Past Course 2 exam problems on options pricing have been difficult, and many of the problems below are adapted from the exams. Similar problems may be asked on the CAS transition exam.}
Exercise 23.1: Black-Scholes Pricing
(Adapted from question 19 of the Fall 2000 Course 2 examination)
A stocks price variance rate, or σ^{2}, 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.
What are the values of the five input parameters to the Black-Scholes model?
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?
What are the values of d_{1} and d_{2}?
What are the values of N(d_{1}), N(–d_{1}), N(d_{2}), and N(–d_{2})?
What is the value of the European call option?
What is the value of the European put option?
Verify that the put call parity relation holds.
Solution 23.1:
Part A: We determine the Black-Scholes parameters:
σ^{2} = 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 d_{1} and d_{2} are
d_{1} = (-0.057 + ½ × 0.25 × 0.25 ] / (0.5 × 0.5) = -0.104
d_{2} = –0.104 – 0.5 × 0.5 = -0.354
Part D: N(d_{1}) = N(–0.104) = 0.459; N(–d_{1}) = N(0.104) = 0.541
N(d_{2}) = N(–0.354) = 0.362; N(–d_{2}) = 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
(Adapted from question 40 of the Spring 2001 Course 2 examination)
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.
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.
What are the values of d_{1} and d_{2}?
What are the values of N(d_{1}), N(–d_{1}), N(d_{2}), and N(–d_{2})?
What is the value of the European call option?
What is the value of the European put option?
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 A PV(X) = S / e^{–0.08} = S × e^{0.08} = $108.33
Part B: The values of d_{1} and d_{2} are
d_{1} = (–0.08 + ½ × 0.16 × 0.25 ] / (0.4 × 0.5) = -0.300
d_{2} = –0.300 – 0.4 × 0.5 = -0.500
Part C: The values are
N(d_{1}) = N(–0.300) = 0.382; N(–d_{1}) = N(0.300) = 0.618
N(d_{2}) = N(–0.500) = 0.309; N(–d_{2}) = 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