Constructing Binomial Trees with Discrete Dividends: 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 31 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. See Section 30 here.

When dividends are paid on a stock on a discrete rather than on a continuous basis, we can use Schroder's method for constructing binomial trees.

The up and down movements (u and d) of the stock price S can be modeled as follows:
u = e^{rh + σ√(h)}

d = e^{rh - σ√(h)}

The discrete dividend is, however, taken into account in calculating the stock price at each of the nodes of the binomial tree. The formula for calculating the stock price is

S_t = F^P_t,T + De^-r(V-t)

The actual stock price volatility in this case is not the same as the prepaid forward price volatility. The relationship between them can be expressed as

σ_F = (S/F^P)σ_S

It is the prepaid forward price volatility that is used in computing u and d in constructing the binomial tree.

The advantage of Schroder's approach is that it produces a recombining binomial tree where, once the nodes have been determined, the option price calculation follows the usual approach in the binomial model.

Meaning of variables:

S = stock price.

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.

T = time to expiration of the option.

V = time at which the dividend is paid, where V < T.

σ_S = annual stock price volatility.

σ_F = annual prepaid forward price volatility.

F^P_t,T = time t prepaid forward price for a forward contract expiring at time T.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 11, pp. 363-365.

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

Problem CBTWDD1. The price of Hideous Co. stock has volatility of 0.45. Hideous Co. stock currently trades for $556 per share, while a prepaid forward contract on Hideous Co. stock currently trades for $523. Find the volatility of the price of a prepaid forward contract on Hideous Co. stock.

Solution CBTWDD1. We use the formula σ_F = (S/F^P)σ_S, where S = 556, F^P = 523, σ_S = 0.45. Thus, σ_F = (556/523)0.45 = σ_F = 0.4783938815

Problem CBTWDD2. The price of Hideous Co. stock has volatility of 0.45. Hideous Co. stock currently trades for $556 per share, while a prepaid forward contract on Hideous Co. stock currently trades for $523. If a dividend of $37 per share is to be paid on Hideous Co. stock in 2 years, find the annual continuously compounded interest rate.

Solution CBTWDD2. We use the formula S_t = F^P_t,T + De^-r(V-t) and rearrange it as

De^-r(V-t) = S_t - F^P_t,T. Here, S₀ = 556, F^P_0,T = 523, V = 2, D = 37. So

37e^-2r = 556 - 523 = 33. Thus, e^-2r = 33/37 and r = -ln(33/37)/2 = r = 0.0572051756

Problem CBTWDD3. The price of Hideous Co. stock has volatility of 0.45. Hideous Co. stock currently trades for $556 per share, while a prepaid forward contract on Hideous Co. stock currently trades for $523. If a dividend of $37 per share is to be paid on Hideous Co. stock in 2 years, find the price of Hideous Co. stock in one year if the stock price declines during that time period. Use a binomial model where one period is equal to one year.

Solution CBTWDD3. We note that the desired σ here is σ_F = 0.4783938815, found in Solution CBTWDD1. Also the desired r here is r = 0.0572051756, found in Solution CBTWDD2. Also, h = 1.

We calculate d = e^{rh - σ√(h)} = e^{0.0572051756 - 0.4783938815} = d = 0.6562662484.

We use the prepaid forward price F^P_0,T = 523 to calculate F^P_1,T = dF^P_0,T = 0.6562662484*523 = 343.2272479

Now we find S₁ = F^P_1,T + De^-r(V-1) = 343.2272479 + 37e^{-0.0572051756(2-1)} = S₁ = $378.1700583

Problem CBTWDD4. Various Industries will pay a dividend of $4 per share in two years. Currently, the stock of Various Industries trades for $69 per share. The annual continuously-compounded interest rate is 0.08, and prepaid forward price volatility on Various Industries stock is 0.05. Find the price of Various Industries stock in one year if the stock goes up. Use a binomial model where one period is equal to one year.

Solution CBTWDD4. First we find the current prepaid forward price: S_t = F^P_t,T + De^-r(V-t) implies that F^P_t,T = S_t -De^-r(V-t), where V = 2, t = 0, r = 0.08, S_t = 69, and D = 4.

Thus, F^P_0,T = 69-4e^-0.08(2) = F^P_0,T = 65.59142484

Now we find u using σ = 0.05 and h = 1:

u = e^{rh + σ√(h)} = e^{0.08 + 0.05} = u = 1.13882838

So the prepaid forward price in one year if the stock goes up will be
F^P_1,T = uF^P_0,T = 1.13882838*65.59142484 = F^P_1,T = 74.69737631.

Now we apply the formula S₁ = F^P_1,T + De^-r(V-1) = 74.69737631 + 4e^-0.08(2-1) = S₁ = $78.3898417

Problem CBTWDD5. Torpid LLC stock currently trades for $2 per share, and stockprice volatility is 0.08. The annual continuously-compounded interest rate is 0.03. Torpid LLC will pay a dividend of $0.5 per share in 1 year. If the stock price of Torpid LLC stock goes down in two months, what will the stock price be? Use a binomial model where one period is equal to two months.

Solution CBTWDD5. First we find the current prepaid forward price: S_t = F^P_t,T + De^-r(V-t) implies that F^P_t,T = S_t -De^-r(V-t), where V = 1, t = 0, r = 0.03, S_t = 2, and D = 0.5.

Thus, F^P_0,T = 2-0.5e^-0.03 = F^P_0,T = 1.514777233.

We note that the desired σ here is σ_F = (S/F^P)σ_S, where S = 2, F^P_0,T = 1.514777233, and σ_S = 0.08. Thus, σ_F = (2/1.514777233)0.08 = σ_F = 0.1056260924

Now we find d using σ = 0.1056260924 and h = 1/6:

d = e^{0.03/6 - 0.1056260924√(1/6)} = e^{-0.0381216717} = d = 0.9625958131

So the prepaid forward price in 2 months if the stock goes down will be
F^P_1/6,T = 0.9625958131F^P_0,T = 0.9625958131*1.514777233 = F^P_1/6,T =1.458118223

Now we apply the formula S_1/6 = F^P_1/6,T + De^-r(V-1/6) = 1.458118223 + 0.5e^-0.03(5/6) =

S_1/6 = $1.945773179

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