The Black-Scholes Formula Using Prepaid Forward Prices: Practice Problems and Solutions

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 34 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.

It is possible to express the Black-Scholes formula using prepaid forward prices for the stock and for the strike asset.

We note that F^P_0,T(S) = Se^-∂T and F^P_0,T(K) = Ke^-rT.

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)

This formula has two advantages over the standard Black-Scholes formula.

1. The interest rate and dividend yield do not appear explicitly in the formula; the formula requires only four parameters to be known rather than six.

2. This formula allows for calculating option prices for options where the strike asset is something other than cash.

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.