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 0.0569 or 5.69%.

Step 1: Determine the properties of the random variable \(Y_n\)

Given \(Y_n = Y_{n-1} + X_n,\) note that the stock's price on the \(n\)th day is the sum of the stock's price on the \((n-1)\)th day and the random variable \(X_n\). Since the \(X_n\) are independent and identically distributed, their sum will have a mean equal to the sum of their individual means and a variance equal to the sum of their individual variances. Recall that the mean (\(\mu_X\)) of each \(X_n\) is 0, and the variance (\(\sigma_X^2\)) is 1. Therefore, the sum of the means is: $$\mu_{Y_{10} - Y_0} = 10 \times \mu_X = 0$$ And the sum of the variances is: $$\sigma_{Y_{10} - Y_0}^2 = 10 \times \sigma_X^2 = 10$$

Step 2: Calculate the standard deviation of the sum

To calculate the standard deviation (\(\sigma_{Y_{10} - Y_0}\)), take the square root of the variance (\(\sigma_{Y_{10} - Y_0}^2\)). $$\sigma_{Y_{10} - Y_0} = \sqrt{10}$$

Step 3: Transform the random variable to Standard Normal distribution

Let \(Z = \frac{Y_{10} - Y_0 - \mu_{Y_{10} - Y_0}}{\sigma_{Y_{10} - Y_0}}\) be a Standard Normal random variable corresponding to the \(Y_{10} - Y_0\). Then, we want to find the probability that the stock's price exceeds 105 after 10 days, i.e., \(P(Y_{10} - Y_0 > 5)\). Using the transformed variable, we can write it as: $$P\left(Z > \frac{5 - \mu_{Y_{10} - Y_0}}{\sigma_{Y_{10} - Y_0}}\right)$$

Step 4: Calculate the probability

Plug in the values of \(\mu_{Y_{10} - Y_0}\) and \(\sigma_{Y_{10} - Y_0}\): $$P\left(Z > \frac{5 - 0}{\sqrt{10}}\right)$$ Now we can use a Standard Normal table or a calculator to find the corresponding probability: $$P(Z > 1.581) \approx 0.0569$$ The probability that the stock's price will exceed 105 after 10 days is approximately 0.0569 or 5.69%.

Key Concepts

Understanding Random Variables

In the context of stock prices and market analysis, the concept of random variables plays a pivotal role. A random variable is a numerical outcome of a random phenomenon. For instance, the daily change in a company's stock price can be modeled as a random variable. This is because the daily fluctuation is uncertain and can vary due to countless unpredictable factors such as market sentiment, economic reports, and global events.

When we say a random variable has a mean (average) of 0, it suggests that the expected long-term gain or loss is zero. This implies that the stock price is as likely to go up as it is to go down on any given day. Variance, denoted by \(\text{\(\sigma^2\)}\), measures the spread or the average of the squared differences from the mean. A low variance indicates that the stock prices are clustered closely around the mean, while a high variance signifies that the stock prices are more spread out.

Modeling stock price changes as random variables helps analysts in making informed predictions and assessing risks associated with stock trading. Still, it's important to remember that these models are simplifications and cannot account for all market complexities.

The Standard Normal Distribution

The standard normal distribution, also known as the Z-distribution, is a specific type of normal distribution with a mean \(\mu\) of 0 and a standard deviation \(\sigma\) of 1. It is a bell-shaped curve that is symmetric about the mean and shows the distribution of a random variable in terms of standard deviations from the mean.

When individual random variables (like daily stock price changes) are independent and identically distributed, as they are in the exercise, their sum can be transformed into a standard normal variable. This is done through a process called standardization, which involves subtracting the mean and dividing by the standard deviation.

Once a random variable is standardized, we can use standard normal distribution tables or software to calculate probabilities for different scenarios. These probabilities help investors understand potential outcomes and make decisions accordingly. For example, in the exercise, we standardized the stock price change over 10 days to determine the likelihood of it exceeding a certain value.

Variance and Standard Deviation

The terms variance and standard deviation are key to understanding the spread or volatility in data, such as stock price movements. Variance measures the average squared difference from the mean, providing a sense of how much variability exists in a set of values. In our stock price scenario, it signifies the uncertainty or risk associated with the fluctuations in price.

To calculate variance, square the differences between each value and the mean and find their average. However, because variance gives us a squared unit, we often use standard deviation, which is the square root of variance, to align the units with the original data.

In the context of the exercise, knowing that the variance is 1 means that we can expect the stock price to deviate from the previous day's price with a standard deviation of \(\text{\(\sqrt{1}\)}\), which equals 1. Over 10 days, the exercise shows us that the variance accumulates, necessitating a recalculation of standard deviation for the 10-day period, which turns out to be \(\text{\(\sqrt{10}\)}\). These concepts enable us to quantify and manage the uncertainty inherent in financial markets.