Multi-Period Binomial Option Pricing with Recombining Trees

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 18

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 18 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.

A recombining binomial option price tree is one in which an up move in the stock price for one period followed by a down move in the stock price in the next period is identical in its result (S_ud) to a down move in the first period followed by an up move in the next (S_du).

In the recombining tree, there are three possible stock prices after two time periods:
S_uu = u²S

S_ud = S_du = udS

S_dd = d²S

We recall from Section 17 that the terms u and d can be found as follows:

u = e^{(r-∂)h + σ√(h)}

d = e^{(r-∂)h - σ√(h)}

Definitions of variables:

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= current 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,

σ = the annualized standard deviation of the continuously compounded stock return.

The way to approach multi-period binomial option pricing models is to figure out the stock and option prices in the latest period and work backward from there using any and all the formulas introduced in Sections 15 through 18.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, pp. 323-328.

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