What Is the Binomial Option Pricing Model?

The binomial option pricing model is an options valuation method developed in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option's expiration date.

The model reduces possibilities of price changes and removes the possibility for arbitrage. A simplified example of a binomial tree might look something like this:

Binomial Option Pricing Model

Another option pricing model is the Black Scholes Model, which makes assumptions about European stock price distribution.

Basics of the Binomial Option Pricing Model

With binomial option price models, the assumptions are that there are two possible outcomes, hence the binomial part of the model. With a pricing model, the two outcomes are a move up, or a move down.

The major advantage to a binomial option pricing model is that they’re mathematically simple. Yet these models can become complex in a multi-period model.

Key Takeaways

  • The binomial option pricing model values options using an iterative approach.
  • With the model, there are two possible outcomes with each iteration—a move up or a move down.
  • It reduces possibilities of price changes, while removing the possibility for arbitrage.
  • The model is mathematically simple.

Real World Example of Binomial Option Pricing Model

A simplified example of a binomial tree has only one step. Assume there is a stock that is priced at $100 per share. In one month, the price of this stock will go up by $10 or go down by $10, creating this situation:

  • Stock price = $100
  • Stock price in one month (up state) = $110
  • Stock price in one month (down state) = $90

Next, assume there is a call option available on this stock that expires in one month and has a strike price of $100. In the up state, this call option is worth $10, and in the down state, it is worth $0. The binomial model can calculate what the price of the call option should be today.

For simplification purposes, assume that an investor purchases one-half share of stock and writes or sells one call option. The total investment today is the price of half a share less the price of the option, and the possible payoffs at the end of the month are:

  • Cost today = $50 - option price
  • Portfolio value (up state) = $55 - max ($110 - $100, 0) = $45
  • Portfolio value (down state) = $45 - max($90 - $100, 0) = $45

The portfolio payoff is equal no matter how the stock price moves. Given this outcome, assuming no arbitrage opportunities, an investor should earn the risk-free rate over the course of the month. The cost today must be equal to the payoff discounted at the risk-free rate for one month. The equation to solve is thus:

  • Option price = $50 - $45 x e ^ (-risk-free rate x T), where e is the mathematical constant 2.7183.

Assuming the risk-free rate is 3% per year, and T equals 0.0833 (one divided by 12), then the price of the call option today is $5.11.

Due to its simple and iterative structure, the binomial option pricing model presents certain unique advantages. For example, since it provides a stream of valuations for a derivative for each node in a span of time, it is useful for valuing derivatives such as American options – which can be executed anytime between the purchase date and expiration date. It is also much simpler than other pricing models such as the Black-Scholes model.