Question

Suppose that stock prices follow a binomial tree, the possible values of \(S(2)\) being \(\$ 121, \$ 110\) and \(\$ 100\). Find \(u\) and \(d\) when \(S(0)=100\) dollars. Do the same when \(S(0)=104\) dollars.

Short answer

When \(S(0) = 100\), \(u = 1.1\) and \(d = 1\).When \(S(0) = 104\), \(u = 1.08\) and \(d = 0.98\).

Step-by-step solution

Understand the Problem

Identify the given values: Possible values of stock prices at time 2, \(S(2)\), are \(121, \)110, and \(100. Also, the initial stock price at time 0, \(S(0)\), is given as \)100 and $104 for separate cases.

Understand the Binomial Tree Model

In a binomial tree model, stock price \(S(T)\) at time \(T\) can take multiple paths. The factor by which the stock price goes up is denoted as \(u\), and the factor by which it goes down is denoted as \(d\).

Determine \(u\) and \(d\) for \(S(0) = 100\)

Given \(S(0) = 100\), and possible values \(S(2) = 121, 110, 100\), create equations based on these possibilities. If \(u\) and \(d\) are respectively the up and down factors, stock prices at \(S(2)\) could be: \(S(0) \times u^2 = 121\), \(S(0) \times u \times d = 110\), and \(S(0) \times d^2 = 100\). Substituting \(S(0) = 100\) in the equations, we get: 1) \(100u^2 = 121\) 2) \(100ud = 110\)3) \(100d^2 = 100\)

Solve for \(u\) and \(d\) for \(S(0) = 100\)

From equation 3: \(d^2 = 1\), so \(d = 1\).From equation 1: \(u^2 = 1.21\), so \(u = 1.1\).Double-check with equation 2: \(100 \times 1.1 \times 1 = 110\), which holds true.

Determine \(u\) and \(d\) for \(S(0) = 104\)

Repeat the same steps, now with \(S(0) = 104\).Equations are based on \(S(2) = 121, 110, 100\):1) \(104u^2 = 121\)2) \(104ud = 110\)3) \(104d^2 = 100\)

Solve for \(u\) and \(d\) for \(S(0) = 104\)

From equation 3: \(d^2 = 0.9615\), so \(d = 0.98\) (taking positive root).From equation 1: \(u^2 = 1.1635\), so \(u = 1.08\) (taking positive root).Verify with equation 2: \(104 \times 1.08 \times 0.98 = 110\), which holds true.

Key concepts

stock price modeling

Stock price modeling is an essential concept in finance. It involves predicting future stock prices based on historical data and mathematical models. One popular method is the binomial tree model. This model takes into account various potential outcomes and calculates the probability of each, making it a powerful tool for pricing options and other financial derivatives.

Stock prices can go up or down in each time period, creating a tree-like structure of possible future prices. By analyzing this tree, financial analysts can predict potential future prices and make informed decisions.

financial engineering

Financial engineering combines financial theory, mathematical tools, and computer programming to solve complex financial problems. It plays a crucial role in developing new financial products and strategies.

This field utilizes various models, including binomial trees, to price options, manage risk, and optimize investment portfolios. Financial engineers apply these models to create innovative solutions and strategies that enhance financial decision-making. By using binomial trees, they can simulate different scenarios and analyze the impact of various factors on stock prices and investment outcomes.

• Price options and derivatives
• Manage risk
• Optimize portfolios

up and down factors

The up and down factors, denoted as \(u\) and \(d\), are crucial in the binomial tree model. These factors indicate the proportional increase or decrease in stock prices at each step.

To calculate \(u\) and \(d\), you need the possible stock prices at a future time. For example, in the given problem: If \(S(0)\) is \(100\), and possible values at \(S(2)\) are \(121, 110, 100\), we solve for \(u\) and \(d\).

Using the equations:
1. \(S(0) \times u^2 = 121\)
2. \(S(0) \times u \times d = 110\)
3. \(S(0) \times d^2 = 100\)

We find that \(u = 1.1\) and \(d = 1\) for \(S(0) = 100\). For \(S(0) = 104\), we calculate \(u = 1.08\) and \(d = 0.98\). These factors help in creating an accurate binomial tree for stock price modeling.

binomial tree calculation

A binomial tree is a graphical representation of potential future values of stock prices. It's constructed using up and down factors \(u\) and \(d\). Stock price moves up by a factor of \(u\) and down by a factor of \(d\) at each step.

To build a binomial tree, you follow these steps:

• Identify initial stock price \(S(0)\).
• Determine up \(u\) and down \(d\) factors using given future prices.
• Calculate subsequent stock prices at each node by multiplying by \(u\) or \(d\).

For example, with \(S(0) = 100\), \(u = 1.1\), and \(d = 1\):

• Step 1: Initial price \(S(0) = 100\).
• Step 2: Calculate \(S(1)\) up: \(S(1) = 100 \times 1.1 = 110\).
• Step 3: Calculate \(S(1)\) down: \(S(1) = 100 \times 1 = 100\).
• Step 4: Extend tree further for future steps.

Using binomial tree calculation, you can predict end-of-period stock prices and make informed financial decisions.