Constructing Binomial Trees for Option Prices: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 17

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 17 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.

Here we explore a method of constructing binomial trees in the one-period binomial option pricing model.

The formula for a forward price is

F_{t, t+h} = e^(r-∂)hS_t where

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

∂ = annual continuously-compounded dividend yield.

F_{t, t+h} = price of forward contract made at time t and expiring at time t + h.

h = one time period in the binomial model.

S_t = stock price at time t.

Furthermore, we let

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

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

σ = the annualized standard deviation of the continuously compounded stock return.

Then the possible evolution of future stock prices can be modeled via the following formulas:
uS_t = F_{t, t+h}e^σ√(h)

dS_t = F_{t, t+h}e^-σ√(h)

The terms u and d can be found as follows:

u = e^{(r-∂)h + σ√(h)}

d = e^{(r-∂)h - σ√(h)}

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, pp. 321-322.

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

Problem CPTOP1. The annualized standard deviation of the continuously compounded stock return for Malicious Co. is currently 0.90. The annual continuously compounded interest rate is 0.07, and Malicious Co. pays dividends on its stock at an annual continuously compounded yield of 0.05. Using the one-period binomial option pricing model, what is the factor by which the price of Malicious Co. might increase in 3 years?