**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 = eσ√(h)

d = e-σ√(h)

This model breaks down if h is too large or σ is too small, such that erh > eσ√(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 = e2σ√(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.

**Solution ABT1.** We use the formulas u = eσ√(h) and d = e-σ√(h), where h = 1/6 and c = 0.3. Thus, u = e0.3√(1/6) = 1.130290283, and d = e-0.3√(1/6) = 0.8847284766. Thus, Suud = 1.130290283*1.130290283*0.8847284766*1230 = **$1390.257048**

**Problem ABT2.** During 54 periods in a binomial model, the stock price of Imperious LLC has gone up 33 times and gone down 21 times. The current price of Imperious LLC stock is $32/share. The stock price volatility is 0.2, and one time period in the binomial model is 6 months. Using a Cox-Rubinstein binomial tree, calculate the original price of Imperious LLC stock.

**Solution ABT2.** We note that with a Cox-Rubinstein binomial tree, ud = eσ√(h)e-σ√(h) = 1, so going up 33 times and down 21 times is equivalent to going up 12 times. u12 = e12σ√(h) = e12*0.2√(1/2) = 5.457857289. The current price is 32, so the price 54 periods ago was 32/5.457857289 = **$5.863106766.**

**Problem ABT3.** During the course of 34 time periods of 1 year each in the binomial model, two stocks that were identically priced at the beginning diverged in their prices. Stock A went up consistently, while Stock B went down consistently. The volatility of both stocks’ prices is 0.09. Find the ratio of the price of Stock A to the price of Stock B.

**Solution ABT3.** We use the formula u/d = e2σ√(h), noting that we seek u34/d34 = (u/d)34 = e68σ√(h) = e68*0.09√(1) = **u34/d34 = 454.8646945**

**Problem ABT4.** The stock prices of Furious LLC can be modeled via a lognormal tree with 6 years as one time period. The annual continuously compounded risk-free interest rate is 0.08, and the stock pays dividends with an annual continuously compounded yield of 0.01. The stock price volatility is 0.43. The stock price is currently $454 per share. Find the stock price in 72 years if it goes up 7 times and down 5 times.

**Solution ABT4.** We use the formulas

u = e(r- δ-0.5σ^2)h + σ√(h) and d = e(r- δ-0.5σ^2)h – σ√(h)

Here, r = 0.08, ∂ = 0.01, h = 6, S = 454, σ = 0.43. Thus,

u = e(0.08- 0.01 -0.5*0.43^2)6 + 0.43√(6) = u = 2.505731203

d = e(0.08- 0.01 -0.5*0.43^2)6 – 0.43√(6) = d = 0.3048362324

Thus, our desired price is u7d5S = 2.50573120370.30483623245*454 = **$741.1910185**

**Problem ABT5.** The stock prices of Obsequious Co. can be modeled via a lognormal tree with 1 day as one time period. The annual continuously compounded risk-free interest rate is 0.4, and the stock pays dividends with an annual continuously compounded yield of 0.2. The stock price volatility is 0.90. Today, Obsequious Co.’s stock price is $9000/share. During the past year, the price went up 43 times and down 322 times. Find the price of Obsequious Co.’s stock one non-leap year ago.

**Solution ABT5.** We use the formulas

u = e(r- δ-0.5σ^2)h + σ√(h) and d = e(r- δ-0.5σ^2)h – σ√(h)

Here, r = 0.4, ∂ = 0.2, h = 1/365, σ = 0.9. Thus,

u = e(0.4- 0.2- 0.5*0.9^2)/365 + 0.9√(1/365) = 1.047646803

d = e(0.4- 0.2- 0.5*0.9^2)/365 – 0.9√(1/365) = 0.9534485668

Using these values for u and d, d322u43 = 0.00000159573726

Thus, the stock price one year ago was 9000/0.00000159573726 = **$5,640,026,227 per share.** (Obsequious Co. stock *really* took a hit over the past year!)

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