Breaking Down Binomial Trees

In the financial world, the Black-Scholes and the binomial option models of valuation are two of the most important concepts in modern financial theory. To value an option, one can use either the Black-Scholes model or the binomial option model, and each has its own advantages and disadvantages. (For a grasp of the building blocks of options, read our Options Basics tutorial.)

Some of the basic advantages of using the binomial model are:

• multiple-period view
• transparency
• ability to incorporate probabilities

In this article, we'll explore the advantages of using the binomial model instead of the Black-Scholes, provide some basic steps to develop the model and explain how it is used. (To read more about the Black-Scholes model, see Understanding Option Pricing and The "True" Cost of Stock Options.)

Multiple-Period View
The binomial model enables a multi-period view of the underlying asset price as well as the price of the option. In contrast to the Black-Scholes model, which provides a numerical result based on five inputs, the binomial model allows for the calculation of the asset and the option for multiple periods along with the range of possible results for each period (see below).

The advantage of this multi-period view is that the user can visualize the change in asset price from period to period and evaluate the option based on making decisions at different points in time. For an American option, which can be exercised at any time before the expiration date, the binomial model can provide insight into when exercising the option may look attractive and when it should be held for longer periods. (Learn to recognize an American option by reading How do you tell whether an option is American or European style?)

By looking at the binomial tree of values, one can determine in advance when a decision on exercise may occur. If the option has a positive value, there is the possibility of exercise, whereas if it has a value less than zero, it should he held for longer periods.

Transparency
Closely related to the multi-period review is the ability of the binomial model to provide transparency into the underlying value of the asset and the option as it progresses through time. The Black-Scholes model has five inputs:

1. Risk-free rate
2. Exercise price
3. Current price of asset
4. Time to maturity
5. Implied volatility of the asset price

When these data points are entered into a Black-Scholes model, the model calculates a value for the option, but the impacts of these factors are not revealed on a period-to-period basis. With the binomial model, one can see the change in the underlying asset price from period to period and the corresponding change caused in the option price. (Read more in Dividends, Interest Rates And Their Effect On Stock Options.)

Incorporating Probabilities
The basic method of calculating the binomial option model is to use the same probability for success and failure each period until option expiration. However, one can actually incorporate different probabilities for each period based on new information obtained as time passes.

For example, there may be a 50/50 chance that the underlying asset price can increase or decrease by 30% in one period. For the second period, however, the probability that the underlying asset price will increase may grow to 70/30. Let's say we are evaluating an oil well; we are not sure what the value of that oil well is, but there is a 50/50 chance that the price will go up. If oil prices go up in Period 1, making the oil well more valuable, and the market fundamentals now point to continued increases in oil prices, the probability of further appreciation in price may now be 70%. The binomial model allows for this flexibility; the Black-Scholes model does not. (For more, see Find The Right Fit With Probability Distributions.)

Developing The Model
The simplest binomial model will have two expected returns, whose probabilities add up to 100%. In our example, there are two possible outcomes for the oil well at each point in time. A more complex version could have three or more different outcomes, each of which is given a probability of occurrence.

To calculate the returns per period starting from time zero (now), we must make a determination of the value of the underlying asset one period from now. In this example, we will assume the following:

• Price of underlying asset: $500
• Investment required: $1,000
• Risk-free rate: 5%
• Price change each period: 30% up or down

The price of the underlying asset is $500, and in Period 1, it can either be worth $650 or $350. That would be the equivalent of a 30% increase or decrease in one period. Applying the same 30% up-or-down calculation to the next period results in values of $845, $455 or $245 for Period 2. We can continue this process for as many periods as are necessary to reach the expiration of the option.

In our example, there is a 50% chance of appreciation and a 50% chance of depreciation. Using the Period 2 values as an example, this calculates out as ($650x50%) + ($350*50%) = $500, which is the current value of the asset. You can now see that if the probabilities are altered, the expected value of the underlying asset will also change. If the probability should be changed, it can also be changed for each subsequent period and does not necessarily have to remain the same throughout. (Read more about changes in value in Appreciating Depreciation.)

As the example shows, the binomial model can be extended easily to multiple periods. Although the Black-Scholes model can calculate the result of an extended expiration date, the binomial model extends the decision points to multiple periods.

Identifying Go/No-Go Decisions
The calculations embedded in the third portion of the example are beyond the scope this article, but you can see where each valuation point results in either a "go" (value of zero) or a "wait" (value of one) decision. Using a three-period expiration as our example, we could see that the optimal decision in Periods 1 and 2 is to wait another period to evaluate the price of the underlying asset and option.

The binomial model allows the user to visualize the future decision points by looking forward on the binomial tree, whereas the Black-Scholes provides an absolute value for the option but no guidance on whether to exercise the option or not. (For more, see Mapping Out The Stock Options Landscape.)

Uses For The Binomial Model
Besides being used for projects or investments with a high degree of uncertainty, the binomial model can also be used for capital-budgeting and resource-allocation decisions, particularly projects with multiple periods or an embedded option to either continue or abandon at certain points in time.

One simple example is a project that entails drilling for oil. The uncertainty of this type of project arises due to the lack of transparency of whether the land being drilled has any oil at all, the amount of oil that can be drilled if oil is found and the price at which the oil can be sold once extracted. (To learn more about investing in oil, read A Guide To Investing In Oil Markets.)

The binomial option model can assist in making decisions at each point of the oil drilling project. For example, assume we decide to drill, but the oil well will only be profitable if we find enough oil and the price of oil exceeds a certain amount. It will take one full period to determine how much oil we can extract as well as the price of oil at that point in time. After the first period (one year, for example), we can decide based on these two data points whether to continue to drill or abandon the project. These decisions can be continuously made until a point is reached where there is no value to drilling, at which time the well will be abandoned.

Conclusion
There are many other types of projects and capital-budgeting processes that can be optimized by the option-evaluation method, but unfortunately, many decisions are based on the present value of a project without incorporating the wide range of results that is possible in the future.

Both the Black-Scholes model and the binomial model can be used to value options, but while each is best suited to a particular situation, the binomial model has a broader range of applications, is much more intuitive and is easier to use.

For further reading, check out Exotic Options: A Getaway From Ordinary Trading.