A binomial tree is used to predict stock price movements assuming there are two possible outcomes, each of which has a known probability of occurrence.

$$

\begin{array}{}

\text{Binomial Stock Price Tree} \\

\end{array}

$$

$$

\begin{array}

\hline

{} & {} & {} & {} & {} & {} & \text{uuuS} \\

{} & {} & {} & {} & \text{uuS} & \Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} & {} \\

{} & {} & \text{uS} & \Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} & {} & {} & \text{uudS} \\

\text{S} & \Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} & {} & {} & \text{udS} & \Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} & {} \\

{} & {} & \text{dS} & \Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} & {} & {} & \text{uddS} \\

{} & {} & {} & {} & \text{ddS} & \Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} & {} \\

{} & {} & {} & {} & {} & {} & \text{dddS} \\

\end{array}

$$

The diagram above shows a series of Bernoulli trials that depict stock price movement as a binomial random variable. We define only two possible outcomes: The price can either move up or down.

*S* denotes the stock price today. *u *denotes 1 plus the rate of return when the stock moves up. Similarly, *d* denotes 1 plus the rate of return when the stock price moves down.

*uS *represents the price of the stock at the **end** of an “up” period. Similarly, *dS *represents the stock price at the end of a “down” period.

The “up transition probability” is the probability of an “up” move while the “down transition probability” is the probability of a “down “ move.

It’s imperative to note that the tree recombines: *udS* = *duS *

**Example: Binomial Tree**

Suppose the initial stock price is $30, *u *= 1.02, *d* = 1/1.02 and the probability of an “up” move is 0.7. Calculate the stock prices after 2 periods.

$$ \text{uuS} = 1.02^2 * 30 = 31.21 $$

Then, we must factor in the probabilities of consecutive “up” movements:

$$ = 0.7 * 0.7 = 0.49 $$

Therefore,

$$ \text{uus} = $31.21 \text{ with probability } 0.49 $$

Similarly,

$$ \text{uds} = 1.02 * \cfrac{1}{1.02} * 30 = $30 \text{ with a probability of } 0.21 (0.7 * 0.3) $$

$$ \text{dus} =\cfrac {1}{1.02} * 1.02 * 30 = $30 \text{ with a probability of } 0.21 (0.3 * 0.7) $$

$$ \text{dds} =\cfrac {1}{1.02} * \cfrac{1}{1.02} * 30 = $28.83 \text{ with } 0.09 \text{ probability } (0.3 * 0.3) $$

Note to candidates:

$$ \text{dus} = \text{uds} = $30 \text{ since the tree} $$

**recombines**: The order of events does not matter.

QuestionUse the binomial tree above to calculate the stock price after 3 periods comprising 2 consecutive periods of stock price growth followed by a reduction in price.

A. $30 with probability 0.49

B. $30.6 with probability 0.147

C. $30.6 with probability 0.49

SolutionThe correct answer is B.

In short, you have been asked to find the stock price after “uud” i.e.

uudS$$ \text{uudS} = 1.02^2 * \cfrac {1}{1.02} * 30 = $30.6 \text{ with a probability of } 0.147 (0.7 * 0.7 * 0.3) $$

You should also note that just like in an ordinary probability tree, the sum of all probabilities at the end of a period must be 1. For example, the sum of probabilities after 2 periods = 0.49 + 0.21 + 0.21 + 0.09 = 1.

The binomial model is applied when pricing derivatives in finance.

*Reading 9 LOS 9g:*

*Construct a binomial tree to describe stock price movement.*