Breaking Down The Binomial Model To Value An Option

In the financial world, the Black-Scholes and the binomial option valuation models are two of the most important concepts in modern financial theory. Both are used to value an option, and each has its own advantages and disadvantages.

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

• a 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 model and provide some basic steps to develop the model and explain how it is used. 

The binomial model provides 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 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 decisions made at different points in time. For a U.S-based option, which can be exercised at any time before the expiration date, the binomial model can provide insight as to when exercising the option may be advisabe and when it should be held for longer periods. By looking at the binomial tree of values, a trader can determine in advance when a decision on an exercise may occur. If the option has a positive value, there is the possibility of exercise whereas, if the option has a value less than zero, it should be held for longer periods.

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 time progresses. The Black-Scholes model has five inputs:

1. The risk-free rate
2. The exercise price
3. The current price of the asset
4. Time to maturity
5. The 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, a trader can see the change in the underlying asset price from period to period and the corresponding change in the option price. 

The basic method of calculating the binomial option model is to use the same probability each period for success and failure until the option expires. However, a trader can 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 percent in one period. For the second period, however, the probability that the underlying asset price will increase may grow to 70/30. For example, if an investor is evaluating an oil well, that investor is 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 percent. The binomial model allows for this flexibility; the Black-Scholes model does not.

The simplest binomial model will have two expected returns whose probabilities add up to 100 percent. 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 assume the following:

• Price of underlying asset (P) : $500
• Call option exercise price (K) : $600
• Risk-free rate for the period: 1 percent
• Price change each period: 30 percent 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 percent increase or decrease in one period. Since the exercise price of the call options we are holding is $600, if the underlying asset ends up being less than $600, the value of the call option would be zero. On the other hand, if the underlying asset exceeds the exercise price of $600, the value of the call option would be the difference between the price of the underlying asset and the exercise price. The formula for this calculation is [max(P-K),0]. 

$\begin{aligned} &\max{\left[\left(P-K\right),0\right]}\\ \\ &\textbf{where:}\\ &P=\text{Price of underlying asset} \\ &K=\text{Call option exercise price} \\ \end{aligned}$​max[(P−K),0]where:P=Price of underlying assetK=Call option exercise price​

Assume there is a 50 percent chance of going up and a 50 percent chance of going down. Using the Period 1 values as an example, this is calculated as

$\begin{aligned} &\max{\left[\left(\$650-\$600\right),0\right]}*0.5+\max{\left[\left(\$ 350-\$ 600\right),0\right]}*0.5\\ & = \$ 50 * 0.5 + \$ 0 = \$ 25\\ \end{aligned}$​max[($650−$600),0]∗0.5+max[($350−$600),0]∗0.5=$50∗0.5+$0=$25​

To get the current value of the call option we need to discount the $25 in Period 1 back to Period 0, which is

$\$25/\left(1+1\%\right) = \$24.75$$25/(1+1%)=$24.75

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.

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.

In addition to its use as a method for calculating the value of an option, the binomial model can also be used for projects or investments with a high degree of uncertainty, capital-budgeting and resource-allocation decisions, and projects with multiple periods or an embedded option to either continue or abandon the project at certain points in time.

One simple example is a project that entails drilling for oil. The uncertainty of this type of project 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. 

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.

The binomial model gives a more detailed view by allowing multi-period views of the underlying asset price and the price of the option for multiple periods as well as the range of possible results for each period. While both the Black-Scholes model and the binomial model can be used to value options, the binomial model has a broader range of applications, is more intuitive, and is easier to use.