The Black-Scholes Formula: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 33

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 33 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10

here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here.

The Black-Scholes formula for option pricing is valid under the following six assumptions, as stated by R. L. McDonald:

"1. Continuously compounded returns on the stock are normally distributed and independent over time.

"2. The volatility of continuously compounded returns is known and constant.

"3. Future dividends are known, either as a dollar amount or as a fixed dividend yield.

"4. The risk-free interest rate is known and constant.

"5. There are no transaction costs or taxes.

"6. It is possible short-sell costlessly and to borrow at the risk-free rate."

The Black-Scholes formula for the call price is

C(S, K, σ, r, T, ∂) = Se^-∂TN(d₁) - Ke^-rTN(d₂)

where d₁ = [ln(S/K) + (r - ∂ + 0.5σ²)T]/[σ√(T)] and d₂ = d₁ - σ√(T)

The Black-Scholes formula for the put price is

P(S, K, σ, r, T, ∂) = Ke^-rTN(-d₂) - Se^-∂TN(-d₁)

We can also get the put formula via put-call parity:
P(S, K, σ, r, T, ∂) = C(S, K, σ, r, T, ∂) + Ke^-rT - Se^-∂T

Meaning of variables:

S = current stock price.

K = strike price of the option.

C = call option price.

P = put option price.

σ = annual stock price volatility.

r = annual continuously compounded risk-free interest rate.

T = time to expiration.

∂ = annual continuously compounded dividend yield.