How The Binomial Model Works

The binomial model is considered to be the best tool nowadays for determining the future value of American styled stock options. It’s better than the traditional Black-Scholes method because it takes into account the possibility of early exercise by calculating values at multiple points in the option’s future time. It also lets you make more accurate predictions. Here’s how it works to make these calculations.

Constructing A Binomial Tree

The ‘bi‘ in binomial is that same one that you find in bicycle or bi-annual. It means two, and there are two possible directions an option can take; it can gain value or lose value. With the binomial model, we construct a ‘tree’ that either grows forward or shrinks down in time. At each branch (or ‘node’) of this tree, two new values are determined. The magnitude of each up/down movement is determined by the volatility used as one of the inputs into the model.

Back To The Future

Starting with the underlying asset’s value, we plot the tree forward until we reach the expiration date; then we work backwards through induction to calculate the present value.

The starting price is the asset’s underlying value. Next, we determine volatility. Volatility represents the chance in value that the stock will gain or lose as represented by a percentage. We also must calculate by what percentage it will gain or lose value. Once we have these percentages, we can start building our tree.

A Simplified Example

For example, let’s say that you have an option which has $100 as its underlying value. There is a 50% chance it will gain in value and a 50% change it will drop. The percentage of gain or loss is calculated to be 14%.

From this, we branch out in the two directions of gain and loss. If the stock price gains, it will be valued at $114 ($100 X 0.14). If it falls, it will be worth $86 ($100 – 14%). The future value of the call option is at least $14 since that is what you will gain if the option is exercised. The future option value on the loss side would be $0 as the call option will be worthless. We’re only interested in determining call option value here.

The future value gain of $14 is halved (because we said it has a 50% chance of going either way). This gives us $7. Working backwards from this future value, we subtract it from the underlying asset value, and this gives us $93. According to the binomial model, this option’s value is $93.

This is a simplified example, of course. In a real binomial tree, you wouldn’t have one future value, but many future values. There would be one at each point in the tree. You would work your way forward until you reached the option’s expiration date, and then work inductively back to the starting value.

As complex as it might sound here, this is actually a very simple mathematical formula. Just compare it to the Black-Scholes model, which is exceedingly difficult for a layman to understand. The binomial model offers us an accurate and easy-to-use method for determining an option’s value, and this is a life-saver in a world where market trends are constantly changing.