Constructing Binomial Trees for Option Prices: 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 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?

Solution CPTOP1. We use the formula u = e^{(r-∂)h + σ√(h)}, and we are given that h = 3, σ = 0.9, r = 0.07, ∂ = 0.05 (so r - ∂ = 0.02). Thus, u = e^{0.02*3 + 0.9√(3)} = e^1.618845727 =

u = 5.047261035

Problem CPTOP2. 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 lower possible price of Malicious Co. might be multiplied in 5 years? (That is, find the possible ratio of Malicious Co.'s lower price in 5 years to its price today.)

Solution CPTOP2. We use the formula d = e^{(r-∂)h - σ√(h)}, and we are given that h = 5, σ = 0.9, r = 0.07, ∂ = 0.05 (so r - ∂ = 0.02). Thus, d = e^{0.02*5 - 0.9√(5)} = e^-1.91246118 = d =

d = 0.1477163823

Problem CPTOP3. The 10-year forward contract on Auspicious, Inc., stock is currently worth $567. The annualized standard deviation of the continuously compounded stock return for Auspicious, Inc., is currently 0.02. If the price of Auspicious, Inc., increases after 10 years, what will it be using the one-period binomial option pricing model?

Solution CPTOP3. We use the formula uS_t = F_{t, t+h}e^σ√(h), and we are given that h = 5, σ = 0.02, F_{t, t+h} = 567. Thus, uS_t = 567*e^0.02√(5) = uS_t = $592.9325586.

Problem CPTOP4. You think that the price of Suspicious LLC stock will decline in 1 month. Currently, a 1-month forward contract on Suspicious LLC stock sells for $423. The annualized standard deviation of the continuously compounded stock return for Suspicious LLC stock is currently 0.56. Using the one-period binomial option pricing model, what might the lower price of Suspicious LLC stock be in 1 month?

Solution CPTOP4. We use the formula dS_t = F_{t, t+h}e^-σ√(h), and we are given that h = 1/12, σ = 0.56, F_{t, t+h} = 423. Thus, dS_t = 423*e^{-0.56√(1/12)} = dS_t = $359.8596534

Problem CPTOP5. Vicious Co. stock may increase or decline in 1 year under the assumptions of the one-period binomial option pricing model. Vicious Co. pays no dividends on its stock, and the annualized standard deviation of the continuously compounded stock return for Vicious Co. stock is 0.81. A 1-year forward contract on Vicious Co. stock currently sells for $100. Vicious Co. stock currently sells for $90. What is the annual continuously compounded risk-free interest rate?

Solution CPTOP5. We first use the formula dS_t = F_{t, t+h}e^-σ√(h), knowing that σ = 0.81, h = 1, F_{t, t+h} = 100, S_t = 90. Thus, d = F_{t, t+h}e^-σ√(h)/S_t = (100/90)e^-0.81 = 0.4942867402

Furthermore, we apply the formula d = e^{(r-∂)h - σ√(h)}. Thus, because the stock pays no dividends, 0.4942867402 = e^r-0.81. Thus, r = ln(0.4942867402) + 0.81 = r = 0.1053605157

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