The Binomial Model

Binomial Model

A binomial model is based on the idea that, over the next period, the value of an asset will change to one of two possible values (binomial). To construct a binomial model, we need to know the beginning asset value, the size of the two possible changes, and the probability of each of these changes occurring.

One Period Binomial Model

Consider a share of stock currently priced at $30. Suppose also that the size of the possible price changes, and the probability of these changes occurring are as follows:

A one period binomial tree for this stock is shown in Figure 29.1. The beginning stock value of $30 is to the left, and to the right are the two possible paths the stock can take, based on that starting point and the size of an up or down move. If the stock price increases by a factor of 1.333 (a return of 33.3%), it ends up at $40.00; if it falls by a factor of 0.75 (a return of −25%), it ends up at $22.50.

Figure 29.1: One Period Binomial Tree

The probabilities of an up move and a down move are calculated based on the size of the moves, as well as the risk free rate, as:

$R_f$ is the periodically compounded annual risk free rate.

We use the expectations approach to calculate the value of an option from this binomial tree. We do this by:

Calculating the payoff of the option at maturity in both the up move and down move states. Calculating the expected value of the option in one year as the probability weighted average of the payoffs in each state.

Discounting the expected value back to today at the risk free rate.

Worked example: Calculating call option value with a one period binomial tree

Calculate the value today of a one year call option on a stock that has an exercise price of $30. Assume that the periodically compounded (as opposed to continuously compounded) risk free rate is 7%, the current value of the stock is $30, the up move factor is 1.333, and the down move factor is 0.75.

Solution

First, we have to calculate the probabilities:

$$\begin{array}{rl}{\pi }_U& =\frac{1+R_f-D}{U-D}=\frac{1+0.07-0.75}{1.33-0.75}=0.55=55\%\\ {\pi }_D& =1-0.55=0.45=45\%\end{array}$$

Next, determine the option payoffs in each state. If the stock moves up to $40, the call option with an exercise price of $30 will pay off $10. If the stock moves down to $22.50, the call option will be worthless.

The option payoffs are illustrated in the following figure.

Let the stock values for the up move and down move be $S^{+}$ and $S^{-}$ and for the call values, $C^{+}$ and $C^{-}$.

With X = $30

The expected value of the option in one period is:

$$E(C_1)=(\$10\times 0.55)+(\$0\times 0.45)=\$5.50$$

The value of the option today, after discounting at the risk free rate of 7%, is:

$$C=\frac{\$5.50}{1.07}=\$5.14$$

We can use the same framework to value a one period put option on the stock.