# Binomial Option Pricing Model

## 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:

## BREAKING DOWN 'Binomial Option Pricing Model'

The binomial option pricing model assumes a perfectly efficient market. Under this assumption, it is able to provide a mathematical valuation of an option at each point in the timeframe specified. The binomial model takes a risk-neutral approach to valuation and assumes that underlying security prices can only either increase or decrease with time until the option expires worthless.

## Binomial Pricing Example

A simplified example of a binomial tree has only one time 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 (up state) = \$110

Stock Price (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. It is also much simpler than other pricing models such as the Black-Scholes model.

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