How the binomial options pricing model values an option step by step

The Black-Scholes model uses advanced maths. The binomial options pricing model reaches similar answers with simple steps. This page explains how the binomial model works, shows a small example and covers where it is used.

Quick answer

The binomial options pricing model values an option by assuming that, in each short period, the price of the underlying asset can move to one of two values: up or down. These steps form a tree of possible prices. The model works out the option's value at the end of the tree and then works backwards to today. It was set out by Cox, Ross and Rubinstein in 1979.

Key takeaways

• The binomial model assumes prices move up or down in each step.
• The steps form a price tree up to expiry.
• Values are worked out at expiry and then brought back to today.
• It can price American options that allow early exercise.
• With many steps, results come close to Black-Scholes.

What is the binomial options pricing model?

'Binomial' means having two parts. In this model, the price of a share or index can only go up by a set amount or down by a set amount in each time step. Repeating this many times creates a tree of possible prices.

John Cox, Stephen Ross and Mark Rubinstein published the model in 1979. It became popular because it is easy to follow and flexible enough to handle options that the original Black-Scholes formula cannot.

A one-step example

Suppose a share trades at Rs 100. In one period, it can rise to Rs 120 or fall to Rs 90. Consider a call option with a strike price of Rs 100. If the share rises, the call is worth Rs 20. If it falls, the call is worth zero.

To keep it simple, assume interest rates are zero. The model finds a 'risk-neutral' probability of the up move that makes the share's expected value equal its price today: (100 minus 90) divided by (120 minus 90), which is one third. The call value is then one third of Rs 20, or about Rs 6.67.

How does the multi-step tree work?

Real models use many small steps instead of one. At each final point, the option's value is simply its payoff. The model then moves back one step at a time, finding the value at each point from the two points ahead of it.

As the number of steps grows, the result gets closer to the Black-Scholes value for a European option. The binomial method is a step-by-step version of the same core idea.

Why use the binomial model?

Its main strength is handling American options, which can be exercised any time before expiry. At each point in the tree, the model checks whether exercising early is worth more than holding. Black-Scholes, in its basic form, cannot do this.

It can also include dividends and changing conditions more easily. In India, exchange-traded index and stock options are European style, but the binomial method is still widely taught and used for other contracts.

The key point to remember

The binomial model breaks time into small up-or-down steps and works backwards from expiry. It is simple to understand and flexible. Like any model, its output depends on its inputs, especially volatility, and does not guarantee a trading result.

Frequently Asked Questions