Binomial Stock Price Tree

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 go up or come down.

S denotes the stock price today while u denotes 1 plus the rate of return when the stock goes up. Yet, d denotes 1 plus the rate of return when the stock price comes down.

uS represents the price of the stock at the end of an “up” period. On the other hand, 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.

Question

Use 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

Solution

The 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.