One-Period Binomial Option Pricing: Practice Problems and Solutions

This is Section 15 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.

The binomial option pricing model makes the simplified assumption that within any given time period, a stock price can only move up by a discrete amount or down by a discrete amount. No other changes are permitted.

With the original stock price being S, the following "tree" shows that in the next time period, the stock price could either be equal to uS (u = 1 + rate of capital gain on stock) or to dS (d = 1 + rate of capital loss on stock).

S - - - uS

S - - - dS

Now we let C_u be the call option price when the stock price increases and C_d be the call option price when the stock price decreases. Let C be the original call option price. The one-period binomial option "tree" for call option prices is as follows:

C - - - C_u

C - - - C_d

There also exists some replicating portfolio which precisely duplicates the option payoff. According to the law of one price, in the absence of arbitrage opportunities, this replicating portfolio must have the same cost as the equivalent call option. The replicating portfolio consists of ∆ (delta) shares and B in lending, expressible as follows:

∆ = e^-∂h(C_u - C_d)/[S(u-d)], where ∂ is the annual continuously-compounded dividend yield and h is the time period in question.

B = e^-rh(uC_d - dC_u)/(u-d), where r is the annual continuously-compounded interest rate.

So the cost of our option is expressible as follows:
C = ∆S + B or

C = e^-rh{C_u[(e^(r-∂)h -d)/(u-d)] + C_d[(u - e^(r-∂)h)/(u-d)]}

You are well-advised to use the former of these two formulas unless it is absolutely impossible to do so.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, pp. 313-317.

We consider the following practice problems that will enable us to use the binomial pricing model for one time period.

Problem OPBOP1. The stock of Predictable Co. is currently worth $100 per share. In one year, this price can either be $120 or $90. Predictable Co. stock does not pay dividends. The annual continuously compounded risk-free interest rate is 5%. The strike price of a European call option on Predictable Co. stock is $110. Using, the one-period binomial option pricing model, find the price today of one such call option on Predictable Co. stock.

Solution OPBOP1. First, we consider the call option price tree

C - - - C_u

C - - - C_d

In one year, if the stock is worth $120, the call option will be worth C_u = 120 - 110 = 10.

If the stock is worth $90, the call option will be worth C_d = 0 (the option cannot be exercised).

We are given ∂ = 0, r = 0.05, S = 100, h = 1, u = 1.2, and d = 0.9.

Thus, ∆ = e^-∂h(C_u - C_d)/[S(u-d)] = 1(10 - 0)/(100*(1.2-0.9)) = 10/30 = ∆ = 1/3

B = e^-0.05(1.2*0 - 0.9*10)/(1.2-0.9) = -28.53688274

C = ∆S + B = (1/3)100 - 28.53688274 = C = $4.796450598

Problem OPBOP2. Impeccable LLC currently has a stock price of $555 per share. A replicating portfolio for a particular call option on Impeccable LLC stock involves borrowing $56 and buying ¾ of one share. Calculate the price of the call option using the one-period binomial option pricing model.

Solution OPBOP2. We use the formula C = ∆S + B. Here, S = 555, B = -56, and ∆ = ¾. (Note: B is negative if money is being borrowed.)

Thus, C = (3/4)555 - 56 = C = $360.25

Problem OPBOP3. A call option on Reliable, Inc., stock currently trades for $45. The stock itself is worth $900 per share. Using the one-period binomial option pricing model, a replicating portfolio for the call option is equal to buying (1/5) shares of stock and borrowing $X. Calculate X.

Solution OPBOP3. We use the formula C = ∆S + B and rearrange it as follows:

B = C - ∆S. Here, C = 45, S = 900, and ∆ = 1/5. Thus, B = 45 - 900/5 = -135. So $135 is borrowed and X = 135.

Problem OPBOP4. Currently, the annual continuously-compounded interest rate is 0.11. Company Co. stock trades for $23 per share, and the annual continuously-compounded dividend yield on Company Co. stock is 0.05. In two months, Company Co. stock will trade for either $18 per share or $29 per share. The strike price of a European call option on Company Co. stock is $25. Using the one-period binomial option pricing model, find the price today of one such call option on Company Co. stock.

Solution OPBOP4.

First, we consider the call option price tree

C - - - C_u

C - - - C_d

In 2 months, if the stock is worth $29, the call option will be worth C_u = 29 - 23 = 6.

If the stock is worth $18, the call option will be worth C_d = 0 (the option cannot be exercised).

We are given ∂ = 0.05, r = 0.11, S = 23, h = 1/6, u = 29/23, and d = 18/23.

Thus, ∆ = e^-∂h(C_u - C_d)/[S(u-d)] = e^-0.05/6(6 - 0)/[23(29/23-18/23)] = e^-0.05/6(6/11) = 0.5409278778

B = e^-rh(uC_d - dC_u)/(u-d) = e^-0.11/6(-(18/23)6)/(29/23-18/23)) = e^-0.11/6(-108/11) =

-9.639821781

C = ∆S + B = 0.5409278778*23 - 9.639821781 = C = $2.801521709

Problem OPBOP5. The stock of Tractable LLC pays dividends. It is currently worth $65; in one year, it will be worth either $45 or $85. ∆ = 0.45 for a replicating portfolio equivalent to one call option on Tractable LLC stock that has a strike price of $64. The option expires in one year. Using the one-period binomial option pricing model, what is Tractable LLC's annual continuously-compounded dividend yield?

Solution OPBOP5. First, we consider the call option price tree

C - - - C_u

C - - - C_d

In one year, if the stock is worth $85, C_u = 85 - 64 = 21. If the stock is worth $45, C_d = 0.

So (C_u - C_d) = 21. Here, S = 65, h = 1, and (u - d) = 85/65 - 45/65 = 40/65 = 8/13. We want to find ∂.

0.45 = ∆ = e^-∂h(C_u - C_d)/[S(u-d)]

0.45 = e^-∂(21/[65(8/13)])

0.45 = 0.525e^-∂

e^-∂ = 6/7

∂ = -ln(6/7) = ∂ = 0.1541506798

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