Binomial Option Pricing with American Options: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 20

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 20 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.

With American options, it is possible for the option to be exercised early. Thus, to determine the option's price at any given "node" in a binomial tree, it is necessary to compare its value if it is held to expiration to the gain that could be realized upon immediate exercise. The higher of these is the American option price.

So for an American put,

P(S, K, t) = max(K - S, e^-rh[P(uS, K, t+h)p* + P(dS, K, t+h)(1-p*)]), where

p* = (e^(r-∂)h - d)/(u - d)

For an American call,

C(S, K, t) = max(S - K, e^-rh[C(uS, K, t+h)p* + C(dS, K, t+h)(1-p*)]),

Definitions of variables:

r = annual continuously-compounded risk-free interest rate.

∂ = annual continuously-compounded dividend yield.

h = one time period in the binomial model.

t = the time equivalent to some "node" in the binomial model.

S= stock price at time t

K = option strike 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.

P(S, K, t) = price of an American put with strike price K and underlying stock price S.

C(S, K, t) = price of an American call with strike price K and underlying stock price S.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 10, p. 329.