The Black Formula for Pricing Options on Futures Contracts: 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 37 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. See Section 35 here. See Section 36 here.

The Black formula for pricing European options on a futures contract is as follows:
For call options:

C(F, K, σ, r, T, r) = Fe^-rTN(d₁) - Ke^-rTN(d₂), where

d₁ = [ln(F/K) + (0.5σ²)T]/[σ√(T)] and d₂ = d₁ - σ√(T)

For put options:

P(F, K, σ, r, T, r) = Ke^-rTN(-d₂) - Fe^-rTN(-d₁)

Also, put-call parity holds:
P(F, K, σ, r, T, r) = C(F, K, σ, r, T, r) + (K - F)e^-rT

Meaning of variables:

F = futures contract price.

K = strike price of the option.

C = call option price.

P = put option price.

σ = annual futures contract price volatility.

r = annual continuously compounded currency risk-free interest rate.

T = time to expiration.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, pp. 381-382.

Problem BFPOFC1. Futures contracts on superwidgets currently trade for $444 per superwidget. The annual futures contract price volatility is 0.15, and the annual continuously compounded currency risk-free interest rate is 0.03. European put options are written on superwidget futures contracts, with a strike price of $454 and time to expiration of 2 years. Find the value of d₁ in the Black formula for the price of such a put option.