The Black-Scholes Formula for Options on Stocks with Discrete Dividends: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 35

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 35 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. See Section 33 here. See Section 34 here.

When stocks pay discrete dividends, we recall that the following relationship holds between the prepaid forward price and the stock price:
F^P_0,T(S) = S₀ - PV_0,T(Div)

For time to expiration T and annual continuously compounded risk-free interest rate, the following equality still holds with respect to the prepaid forward on the strike asset:

F^P_0,T(K) = Ke^-rT.

Making the substitutions above, we can now use the Black-Scholes formula from Section 34 - the formula that uses prepaid forward prices.

Thus, the Black-Scholes formula for the call price is

C(F^P_0,T(S), F^P_0,T(K), σ, T) = F^P_0,T(S)N(d₁) - F^P_0,T(K)N(d₂)

where d₁ = [ln(F^P_0,T(S)/F^P_0,T(K)) + 0.5σ²T]/[σ√(T)] and d₂ = d₁ - σ√(T)

The Black-Scholes formula for the put price is

P(F^P_0,T(S), F^P_0,T(K), σ, T) = F^P_0,T(K)N(-d₂)- F^P_0,T(S)N(-d₁)

We can also get the put formula via put-call parity:
P(F^P_0,T(S), F^P_0,T(K), σ, T) = C(F^P_0,T(S), F^P_0,T(K), σ, T) + F^P_0,T(K) - F^P_0,T(S)

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.