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 that can be exercised before the expiration date, the binomial model can provide insight into when exercising the option may be advisable and when it should be held for extended periods.

By looking at the binomial tree of values, a trader can determine 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 extended periods.

Calculating Price with the Binomial Model

The primary 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.

A binomial tree is valuable when pricing American and embedded options. Its simplicity is its advantage and disadvantage at the same time. The tree is easy to model mechanically, but the problem lies in the possible values the underlying asset can take over some time. In a binomial tree model, the underlying asset can only be worth precisely one of two possible values, which is unrealistic, as assets can be worth any number of values within any given range.

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. Suppose oil prices go up during the period, making the oil well more valuable, and the market fundamentals now point to continued increases in oil prices. In that case, the probability of further price appreciation may now be 70 percent. The binomial model allows for this flexibility; the Black-Scholes model does not.

Real-World Example of Binomial Option Pricing Model

A simplified example of a binomial tree has only one 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 in one month (upstate) = $110
• Stock price in one month (down state) = $90

Next, assume a call option is available on this stock that expires in one month and has a strike price of $100. This call option is worth $10 in the upstate, and in the downstate, 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 a one-half share of stock and writes or sells a 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 (upstate) = $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 month. Today’s cost must equal 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.

The binomial pricing model presents two advantages for option sellers over the Black-Scholes model. The first is its simplicity, which allows for fewer errors in the commercial application. The second is its iterative operation, which adjusts prices promptly to reduce buyers’ opportunity to execute arbitrage strategies.

For example, since it provides a stream of valuations for a derivative for each node in a period, it helps value derivatives such as American options, which can be executed anytime between the purchase date and expiration date. It is also much more straightforward than other pricing models, such as the Black-Scholes model.