Problem 11 - A First Course in Probability

Question

Many people believe that the daily change of price of a company's stock on the stock market is a random variable with mean 0 and variance \(\sigma^{2}\). That is, if \(Y_{n}\) represents the price of the stock on the \(n\) th day, then $$Y_{n}=Y_{n-1}+X_{n} \quad n \geq 1$$ where \(X_{1}, X_{2}, \ldots\) are independent and identically distributed random variables with mean 0 and variance \(\sigma^{2}\). Suppose that the stock's price today is \(100 .\) If \(\sigma^{2}=1,\) what can you say about the probability that the stock's price will exceed 105 after 10 days?

Step-by-Step Solution

Answer

The probability that the stock's price will exceed 105 after 10 days is approximately 5.7%.

Step 1: Write the stock's price equation

Since the stock's price on the nth day, \(Y_n\), is the price on the previous day, \(Y_{n-1}\), plus the change in price on the nth day, \(X_n\), we can write the equation representing the stock's price as: $$Y_{n}=Y_{n-1}+X_{n} \quad n \geq 1$$

Step 2: Sum the daily changes in price over 10 days

In order to find the stock's price after 10 days, we need to sum the daily changes in price for each of the 10 days. We do this using the definition of \(Y_n\) for 10 days: $$Y_{10} = Y_0 + X_1 + X_2 + \cdots + X_{10}$$ where \(Y_0\) is the stock's price today, which is 100.

Step 3: Find the mean and variance of the sum of the daily changes in price

To calculate the probability of exceeding 105 in 10 days, we first need to find the mean and variance of the sum of the daily changes in price over the 10 days. Since \(X_1, X_2, \ldots, X_{10}\) are independent and identically distributed random variables with mean 0 and variance \(\sigma^2 = 1\), the mean and variance of their sum can be calculated as follows: Mean: $$E(Y_{10} - Y_0) = E(X_1 + X_2 + \cdots + X_{10}) = 0$$ Variance: $$Var(Y_{10} - Y_0) = Var(X_1 + X_2 + \cdots + X_{10}) = 10\sigma^2 = 10$$

Step 4: Calculate the probability that the stock's price will exceed 105

To find the probability that the stock's price will exceed 105 after 10 days, we calculate the probability that the sum of the daily changes in price will be greater than 5 (since the stock's price today is 100 and we are interested in the price exceeding 105). This is given by: $$P(Y_{10} - Y_0 > 5)$$ Since we are dealing with a sum of independent and identically distributed random variables, the distribution of the sum is also a normal distribution. Thus, we can standardize the distribution and use the standard normal table or a calculator to find the probability: $$P\left(\frac{Y_{10} - Y_0 - E(Y_{10} - Y_0)}{\sqrt{Var(Y_{10} - Y_0)}} > \frac{5 - 0}{\sqrt{10}}\right) = P(Z > 1.581)$$ where Z is the standard normal random variable. Finally, we can use the standard normal table or a calculator to find the probability: $$P(Z > 1.581) = 1 - P(Z \le 1.581) \approx 1 - 0.943 = 0.057$$

Step 5: Conclude the solution

Based on our calculations, there is approximately a 5.7% chance that the stock's price will exceed 105 after 10 days.

Key Concepts

Random Variable

In the world of probability and statistics, a random variable is an essential concept that plays a critical role in many applications, including financial modeling and risk assessment. A random variable, often denoted by capital letters like X or Y, is a numerical description of the outcomes of a random phenomenon. It assigns numerical values to each possible outcome of a random process, allowing us to quantify uncertainty and make predictions about future events.

For example, when we talk about the daily change in the price of a company's stock, as seen in the exercise, this change is considered a random variable. It reflects the outcome of various unpredictable factors affecting the stock market, such as economic news or investor sentiment. Random variables can be either discrete, taking on a countable number of distinct values, or continuous, capable of taking on any value within an interval or range.

Variance and Mean

The variance and mean are two fundamental statistical measures that provide insight into the behavior of random variables. The mean, represented by \( E(X) \) or \( \mu \) when talking about expected values, is the average outcome that we would expect from a random variable if we were to observe it many times. It signifies the central tendency or the typical value around which the values of the random variable tend to cluster.

The variance, denoted by \( Var(X) \) or \( \sigma^2 \) for a population and \( s^2 \) for a sample, measures how much the values of a random variable spread out from the mean. A high variance indicates that the random variable's values are widely scattered, while a low variance suggests they are closely bunched together. Understanding both the variance and the mean is crucial for predicting outcomes, such as estimating the likelihood of a stock's price exceeding a certain level after a given period.

Sum of Independent Random Variables

When dealing with multiple independent random variables, we often want to understand the behavior of their combined effect. The sum of independent random variables is itself a random variable, and if these variables are independent and identically distributed (i.i.d.), their sum has a mean and variance that can be easily calculated.

The mean of the sum is the sum of the means, which, if all individual random variables have a mean of zero, makes the mean of the sum also zero. The variance of the sum of independent random variables is the sum of their variances. So, if we have ten i.i.d. random variables each with a variance of \( \sigma^2 = 1 \), the variance of the sum is \( 10 \cdot 1 = 10 \). This principle is used in the exercise to determine the variance after ten days, providing crucial information for the probability of stock price fluctuations.

Normal Distribution

The normal distribution, colloquially known as the 'bell curve', is a continuous probability distribution characterized by its symmetric bell-shaped curve. It is determined by two parameters: the mean \( \mu \) which locates the center of the distribution, and the variance \( \sigma^2 \) which indicates the spread or dispersion around the mean.

Many real-world phenomena tend to have a normal distribution under the Central Limit Theorem, when summing up a large number of independent, identically distributed variables. In the context of stock price movements described in the exercise, we make use of this theorem. The sum of daily stock price changes follows a normal distribution, allowing us to apply statistical techniques to evaluate probabilities, such as the probability of the stock price exceeding a certain threshold after a given time. Applying the standard normal distribution simplifies these calculations and provides the ability to find probabilities using known values from the standard normal table or computational tools.