Binomial Tree: Overview, Examples, and Formulas

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.

How a Binomial Tree Works

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. 

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. A binomial tree allows investors to assess when and if an option will be exercised. An option has a higher probability of being exercised if the option has a positive value. 

Special Considerations

The binomial options pricing model (BOPM) is a method for valuing options. The first step of the BOPM is to build the binomial tree. The BOPM is based on the underlying asset over a period of time versus a single point in time. 

 There are a few major assumptions in a binomial option pricing model. First, there are only two possible prices, one up and one down. Second, the underlying asset pays no dividends. Third, the interest rate is constant, and fourth, there are no taxes and transaction costs.

Binomial Tree vs. Black-Scholes Model

The Black Scholes model is another method for valuing options. Computing the price using the binomial tree is slower than the Black Scholes model. However, the binomial tree and BOPM are more accurate. This is especially true for options that are longer-dated and those securities with dividend payments. 

The Black Scholes model is more reliable when it comes to complicated options and those with lots of uncertainty. When it comes to European options without dividends, the output of the binomial model and Black Scholes model converge as the time steps increase. 

Example of a Binomial Tree

Assume a stock has a 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 $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.