One-Period Binomial Option Pricing: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 15

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 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.