The Black Formula for Pricing Options on Futures Contracts: Practice Problems and Solutions

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.

Solution BFPOFC1. We use the formula d₁ = [ln(F/K) + (0.5σ²)T]/[σ√(T)], where F = 444, K = 454, T = 2, σ = 0.15. Thus, d₁ = [ln(444/454) + (0.5*0.15²)2]/[0.15√(2)] = d₁ = 0.001071806

Problem BFPOFC2. 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 d₂ in the Black formula for the price of such a put option.

Solution BFPOFC2. We use the formula d₂ = d₁ - σ√(T), where d₁ = 0.001071806 from Solution BFPOFC1, T = 2, and σ = 0.15. Thus, d₂ = 0.001071806 - 0.15√(2) = d₂ = -0.21110602284

Problem BFPOFC3. 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. Use the Black formula to find the price of one such put option.

Solution BFPOFC3. We use the formula P(F, K, σ, r, T, r) = Ke^-rTN(-d₂) - Fe^-rTN(-d₁), where it is given that F = 444, K = 454, r = 0.03, T = 2, σ = 0.15. Furthermore, from Solutions BFPOFC1-2 that d₁ = 0.001071806 and d₂ = -0.21110602284.

In MS Excel, using the input "=NormSDist(-0.001071806)", we find that N(-d₁) = 0.499572409

In MS Excel, using the input "=NormSDist(0.21110602284)", we find that N(-d₂) = 0.5835977

Thus, P(F, K, σ, r, T, r) = 454e^-0.03*20.5835977 - 444e^-0.03*20.499572409 = P(F, K, σ, r, T, r) = $40.63074147

Problem BFPOFC4. 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 call options are written on superwidget futures contracts, with a strike price of $454 and time to expiration of 2 years. Find the price of one such call option.

Solution BFPOFC4. We use put-call parity: P(F, K, σ, r, T, r) = C(F, K, σ, r, T, r) + (K - F)e^-rT, rearranging the formula thus: C(F, K, σ, r, T, r) = P(F, K, σ, r, T, r) + (F - K)e^-rT. Since F = 444 and K = 454, (F - K) = -10. We are also given that T = 2, r = 0.03, and, from Solution BFPOFC3, P(F, K, σ, r, T, r) = 40.63074147.

Thus, C(F, K, σ, r, T, r) = 40.63074147 - 10e^-0.03*2 = C(F, K, σ, r, T, r) = 31.21309613

Problem BFPOFC5. The dividend yield in the Black-Scholes formula for stock option pricing is analogous to which of these variables in other related formulas? More than one answer may be correct.

(a) The risk-free interest rate in the Black-Scholes formula for stock option pricing.
(b) The risk-free interest rate in the Black formula for futures contract option pricing.
(c) The domestic risk-free interest rate in the Garman-Kohlhagen formula for currency option pricing.
(d) The foreign risk-free interest rate in the Garman-Kohlhagen formula for currency option pricing.
(e) The volatility in the Black formula for futures contract option pricing.

Solution BFPOFC5. We are looking for equivalents to ∂ in the Black-Scholes formula C(S, K, σ, r, T, ∂) = Se^-∂TN(d₁) - Ke^-rTN(d₂). Clearly, r is not necessarily the same as ∂ in this formula, so (a) is incorrect. In all variants of the Black-Scholes, formula, σ is distinct from ∂, so (e) is incorrect. In the Garman-Kohlhagen formula, C(x, K, σ, r, T, f) = xe^-fTN(d₁) - Ke^-rTN(d₂), the foreign risk-free interest rate f is analogous to ∂, while the domestic interest rate r is used in the manner of r in all other variants of the Black-Scholes formula. Thus, (c) is incorrect and (d) is correct. In the Black formula, C(F, K, σ, r, T, r) = Fe^-rTN(d₁) - Ke^-rTN(d₂), r is used in place of both r and ∂, so (b) is correct. Thus, (b) and (d) are correct answers.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.