Alternative Binomial Trees: 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 30 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.

There are several alternative ways to construct a binomial tree.

The Cox-Rubinstein binomial tree can be constructed using these formulas:

u = e^σ√(h)

d = e^-σ√(h)

This model breaks down if h is too large or σ is too small, such that e^rh > e^σ√(h).

The lognormal tree can be constructed using these formulas:

u = e^{(r- δ-0.5σ^2)h + σ√(h)}

d = e^{(r- δ-0.5σ^2)h - σ√(h)}

All methods of binomial tree construction give the same ratio of u to d:

u/d = e^2σ√(h)

ln(u/d) = 2σ√(h)

Meaning of variables:

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

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

h = one time period in binomial model.

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

∂ = annual continuously-compounded dividend yield.

σ = annual stock price volatility.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 11, pp. 359.

Problem ABT1. The stock price of Particular Co. has volatility of 0.3. The stock currently trades for $1230/share. Using 2 months as one time period and a three-period binomial model, calculate the price of the stock if it goes up twice and down once.