Binomial Tree

What is a 'Binomial Tree'

A binomial tree is a graphical representation of possible intrinsic values that an option may take at different nodes or time periods. The value of the option depends on the underlying stock or bond, and the value of the option at any node depends on the probability that the price of the underlying asset will either decrease or increase at any given node.

BREAKING DOWN 'Binomial Tree'

A binomial tree is a useful tool when pricing American options and embedded options. Its simplicity is its advantage and disadvantage at the same time. The tree is easy to model out mechanically, but the problem lies in the possible values the underlying asset can take in one period time. In a binomial tree model, the underlying asset can only be worth exactly one of two possible values, which is not realistic, as assets can be worth any number of values within any given range.

Example of a Binomial Tree

There are a few major assumptions in a binomial option pricing model: 1) only two possible prices, one up and one down; 2) the underlying asset pays no dividends; 3) interest rate is constant; and 4) no taxes and transaction costs.

Assume a stock price of $100, option strike price of $100, one-year expiration date, and interest rate (r) of 5%. At the end of the year there is a 50% probability the stock will rise to $125 and 50% probability it will drop to to $90. If the stock rises to $125 the value of the option will be $25 ($125 stock price minus $100 strike price) and if it drops to $90 the option will be worthless. The option value will be:

Option value = [(probability of rise*up value) + (probability of drop*down value)] / (1+r) = [(0.50*$25) + (0.50*$0)] / (1+0.05) = $11.90