Binomial Pricing for 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 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: