Question

Suppose that the stock price on any given day can either be \(5 \%\) higher or \(4 \%\) lower than on the previous day. Sketch a tree representing possible stock price movements over the next three days, given that the price today is $$\$ 20$$. How many different scenarios can be distinguished?

Short answer

8 distinct scenarios.

Step-by-step solution

- Initial Price

Start with the given initial stock price of $20.

- Day 1 Possibilities

On the first day, the stock price can either increase by 5% or decrease by 4%. Compute these new prices: Increase: \( 20 \times 1.05 = 21 \) Decrease: \( 20 \times 0.96 = 19.2 \)

- Day 2 Possibilities

For the second day, consider both possible prices from Day 1 and compute the next set of prices: If the price is 21, then: Increase: \( 21 \times 1.05 = 22.05 \) Decrease: \( 21 \times 0.96 = 20.16 \) If the price is 19.2, then: Increase: \( 19.2 \times 1.05 = 20.16 \) Decrease: \( 19.2 \times 0.96 = 18.432 \)

- Day 3 Possibilities

For the third day, consider each price from Day 2 and compute the next set of prices: If the price is 22.05, then: Increase: \( 22.05 \times 1.05 = 23.1525 \) Decrease: \( 22.05 \times 0.96 = 21.168 \) If the price is 20.16, then: Increase: \( 20.16 \times 1.05 = 21.168 \) Decrease: \( 20.16 \times 0.96 = 19.3536 \) If the price is 18.432, then: Increase: \( 18.432 \times 1.05 = 19.3536 \) Decrease: \( 18.432 \times 0.96 = 17.69472 \)

- Tree Construction and Counting Scenarios

The tree has branches splitting at each day's prices. Branches expand to represent the price changes for three days: Day 1: 2 possibilities (21, 19.2) Day 2: 4 possibilities (22.05, 20.16, 20.16, 18.432) Day 3: 8 possibilities (23.1525, 21.168, 21.168, 19.3536, 21.168, 19.3536, 19.3536, 17.69472) There are a total of 8 distinct possible scenarios.

Key concepts

stock price movements

Stock price movements refer to the variations in the price of a stock over time. These movements can be caused by a variety of factors including market demand, investor sentiment, economic indicators, and company performance. In the context of our exercise, we assume the stock price can change in a deterministic way each day, either increasing by 5% or decreasing by 4%.
Calculating potential prices over multiple days helps us understand the range of scenarios that could unfold. For example, starting at a price of \(20:

• On Day 1, if the price increases by 5%, we get \)21; if it decreases by 4%, we get $19.20.
• On Day 2, each of those prices can again increase by 5% or decrease by 4%, generating new potential prices.
• Continuing this process over multiple days helps us construct a “tree” of possible prices.

This tree structure allows us to see all potential outcomes based on the initial price and probabilistic changes at each step.

binomial tree model

The binomial tree model is a mathematical model used to predict stock price movements over time. It uses a discrete-time framework for valuing options and other financial derivatives. The model involves creating a tree where each node represents a possible price of the stock at a given point in the future.
The key steps in constructing a binomial tree for stock prices include:

• Starting with the current stock price at the root of the tree.
• Creating branches to represent possible future price movements (e.g., an increase of 5% or decrease of 4%).
• Continuing this branching structure over the desired number of time periods (e.g., days or months).

In our exercise, each day offers two possible price movements, forming a two-branch tree. Over three days, the tree grows to have multiple layers, each representing a specific day. This way, we can visualize the 8 unique scenarios based on increases and decreases in stock price over time. This model is highly valuable for financial analysts and investors as it helps predict future stock prices under defined probabilistic rules.

probabilistic scenarios

Probabilistic scenarios involve calculating different possible outcomes based on the probabilities of certain events occurring. In the context of our stock price movements, we consider how probable changes in price affect potential future prices.
A key part of understanding probabilistic scenarios is considering the likelihood of each outcome. For example:

• If the stock has a 50% chance of increasing by 5% and a 50% chance of decreasing by 4% each day, we map out all possible paths the stock price can take.
• By using a binomial tree model, we trace each path over three days to see where these probabilistic changes lead us.
• Each branch of the tree represents a different scenario, and together they show the complete set of possible futures for the stock price.

By considering these scenarios, investors can better manage risk and make more informed decisions about buying, holding, or selling stocks. Understanding these scenarios assists in predictability and strategic planning based on likely outcomes rather than mere speculation.