VRM15 - The Black-Scholes-Merton Model

Question 1

Explain the lognormal property of stock prices, the distribution of rates of return, and the calculation of expected return

Answer 1

Stock price returns tend to show lognormal distribution

Question 2

Describe the assumptions underlying the Black-Scholes-Merton option pricing model

Answer 2

• mu and sigma constant
• no transaction costs / taxes
• securities trading is continuous
• rf constant
• no dividends or americans
• no riskless arbitrage
• borrow and lend at rf

Question 3

Compute the value of a EU option using the Black-Scholes-Merton model

Answer 3

c = S0 * exp(-qT) * N(d1) - K * exp(-rT) * N(d2)
p = K * exp(-rT) * N(-d2) - S0 * exp(-qT) * N(-d1)
d1 = (ln(s0 / K) + (r - q + sigma^2 / 2) * T) / (sigma * sqrt(T))
d2 = d1 - sigma*sqrt(T)

S0 = stock price now
q = dividend rate / foreign risk free rate (set to zero if no div or home currency)
r = risk free rate

If a discrete dividend, replace S with (S-D) where D is PV discounted continously of the dividends over T

Question 4

Explain how dividends affect the decision to exercise early for American call and put options

Answer 4

Can sometimes be optimal to exercise right before ex-div date

Question 5

Describe warrants, calculate the value of a warrant, and calculate the dilution cost of the warrant to existing shareholders

Answer 5

Price of warrant is calculated using BSM using share price just before announced

Warrants are options issued by a company on its own stock - if exercised dilutes company as it issues more shares

N / N+M * price of warrant is price to existing shareholders