Binomial Pricing for Options on Futures Contracts: Practice Problems and Solutions

This is Section 22 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.

This use of the binomial model to price options on futures contracts assumes that the futures price is equal to the forward price. If this is the case, then the following formulas apply.

u = e^σ√(h)

d= e^-σ√(h)

Δ(dF-F) + Be^rh = C_d

Δ(uF-F) + Be^rh = C_u

Δ = (C_u - C_d)/[F(u-d)]

B = e^-rh[C_u(1-d)/(u-d) + C_d(u-1)/(u-d)]; B is also the value of the option.

p* = (1-d)/(u-d)

Definitions of variables:

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

u = 1 + rate of capital gain on stock if futures price increases.

d = 1 + rate of capital loss on stock if futures price decreases.

σ = the annualized standard deviation of the continuously compounded return on the futures contract.

p* = the risk-neutral probability of an increase in the futures price.

h = one time period in the binomial model.

∆ (delta) = the number of units of the futures contract contained in the replicating portfolio for the option.

B = the number of dollars lent out in the replicating portfolio for the option - equivalent to the option value for options on futures contracts.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, p. 333-334.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem BPOFC1. Futures Contract Ω sells for $90 today. The annualized standard deviation of the continuously compounded return on the futures contract is 0.34. Using a three-period binomial option pricing model, find the price of Futures Contract Ω after 6 years if the contract always increases in price every time period.

Solution BPOFC1. We use the formula u = e^σ√(h), where h = 2 and σ = 0.34. Thus, u = e^0.34√(2) = u = 1.617420524.

So, after three periods, during each of which Futures Contract Ω goes up in price, the futures contract price will be F_uuu = 1.617420524³*90 = F_uuu = 380.8126432

Problem BPOFC2. Futures Contract Ω sells for $90 today. The annualized standard deviation of the continuously compounded return on the futures contract is 0.34. Using a binomial option pricing model, find the risk-neutral probability that the price of Futures Contract Ω will increase during any given two-year period.

Solution BPOFC2. We are given that h = 2 and σ = 0.34.

We use the formula p* = (1-d)/(u-d), knowing from Solution BPOFC1 that

u = 1.617420524. To find d, we use the formula d = e^-σ√(h) = e^-0.34√(2) = d = 0.6182684002

Thus, p* = (1-0.6182684002)/(1.617420524 - 0.6182684002) = p* = 0.3820555355

Problem BPOFC3. Futures Contract Ω sells for $90 today. The annualized standard deviation of the continuously compounded return on the futures contract is 0.34. The annual continuously compounded risk-free interest rate is 0.05. Using a three-period binomial option pricing model, find the price of one six-year European call option on Futures Contract Ω with a strike price of $80.

Solution BPOFC3. We are given that h = 2, r = 0.05, and σ = 0.34.

We assume a recombining binomial tree.

We know from Solutions BPOFC1-2 that u = 1.617420524 and d = 0.6182684002.

We already know from Solution BPOFC1 that

F_uuu = 380.8126432 and thus C_uuu = 300.8126432

F_duu = 1.617420524²*0.6182684002*90 = 145.5678472 and thus C_duu = 65.56784718

F_ddu = 1.617420524*0.6182684002²*90 = 55.64415602 and thus C_ddu = 0

Naturally, F_ddd< F_ddu, so C_ddd = 0 as well.

We know from Solution BPOFC2 that p* = 0.3820555355

For a three-period binomial model, we apply the formula

C = e^-3rh[(p*)³C_uuu + 3(p*)²(1-p*)C_duu + 3(p*)(1-p*)²C_ddu + (1-p*)³C_ddd].

Thus,

C = e^-3*0.05*2[(0.3820555355)³300.8126432 + 3(0.3820555355)²(1-0.3820555355)65.56784718] = C = $25.57156033

Problem BPOFC4. Futures Contract Ω sells for $90 today. The annualized standard deviation of the continuously compounded return on the futures contract is 0.34. The annual continuously compounded risk-free interest rate is 0.05. Using a one-period binomial option pricing model, find ∆ for a replicating portfolio equivalent to one two-year European call option on Futures Contract Ω with a strike price of $30.

Solution BPOFC4. We are given that h = 2, r = 0.05, F = 90, and σ = 0.34.

We know from Solutions BPOFC1-2 that u = 1.617420524 and d = 0.6182684002.

F_u = 1.617420524*90 = 145.5678472 and thus C_u = 115.5678472

F_d = 0.6182684002*90 = 55.64415602 and thus C_d = 55.64415602

We use the formula Δ = (C_u - C_d)/[F(u-d)] =

(115.5678472 - 55.64415602)/[90(1.617420524 - 0.6182684002)] = Δ = 0.6663838017

Problem BPOFC5. Futures Contract Ξ sells for $49 today. The annual continuously compounded risk-free interest rate is 0.15. The price today of one particular three-month European call option on Futures Contract Ξ is $10. ∆ for a replicating portfolio equivalent to one such option is 0.4. If in three months, Futures Contract Ξ will be worth 0.85 of its present amount, what will the price of the call option be? Use a one-period binomial option pricing model.

Solution BPOFC5. We are asked to find Δ(dF-F) + Be^rh = C_d, given that F = 49, d = 0.85, Δ = 0.4, B = 10, r = 0.15, and h = 0.25.

Thus, Δ(dF-F) + Be^rh = 0.4(0.85*49 - 49) + 10e^0.15*0.25 = C_d = $7.442119971

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