The Black-Scholes Formula Using Prepaid Forward Prices: Practice Problems and Solutions
The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 34
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
It is possible to express the Black-Scholes formula using prepaid forward prices for the stock and for the strike asset.
We note that FP0,T(S) = Se-∂T and FP0,T(K) = Ke-rT.
Thus, the Black-Scholes formula for the call price is
C(FP0,T(S), FP0,T(K), σ, T) = FP0,T(S)N(d1) - FP0,T(K)N(d2)
where d1 = [ln(FP0,T(S)/FP0,T(K)) + 0.5σ2T]/[σ√(T)] and d2 = d1 - σ√(T)
The Black-Scholes formula for the put price is
P(FP0,T(S), FP0,T(K), σ, T) = FP0,T(K)N(-d2)- FP0,T(S)N(-d1)
We can also get the put formula via put-call parity:
P(FP0,T(S), FP0,T(K), σ, T) = C(FP0,T(S), FP0,T(K), σ, T) + FP0,T(K) - FP0,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.
Related information
This version of the Black-Scholes formula allows for calculating option prices for options where the strike asset is something other than cash.
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Posted on 02/24/2009 at 9:02:48 AM