Problem 29 - A First Course in Probability

Question

A model for the movement of a stock supposes that if the present price of the stock is \(s,\) then after one period, it will be either \(u s\) with probability \(p\) or \(d s\) with probability \(1-p .\) Assuming that successive movements are independent, approximate the probability that the stock's price will be up at least 30 percent after the next 1000 periods if \(u=1.012, d=0.990,\) and \(p=.52\).

Step-by-Step Solution

Answer

Approximately, the probability that the stock's price will be up at least 30 percent after the next 1000 periods is \(97.95\%\).

Step 1: Determine the minimum number of increases to achieve a 30% gain

Find the smallest value of \(n\), the number of times the stock needs to increase, such that \(u^nds^{1000-n}\geq1.3\). Using the given values of \(u\) and \(d\), we need to solve for \(n\): \[(1.012)^n (0.990)^{1000-n} \geq 1.3\]

Step 2: Find the maximum integer value of n that satisfies the inequality

Solve the inequality by finding the maximum integer value of \(n\), which can be done by logarithms. Divide both sides of the inequality by \((0.990)^{1000}\): \[\frac{(1.012)^n}{(0.990)^n} \geq \frac{1.3}{(0.990)^{1000}}\] Taking the logarithm of both sides: \[n \cdot \log{\frac{1.012}{0.990}} \geq \log{\frac{1.3}{(0.990)^{1000}}}\] Now, divide by \(\log{\frac{1.012}{0.990}}\) and round down to the nearest integer to find the maximum value of \(n\): \[n \geq \frac{\log{\frac{1.3}{(0.990)^{1000}}}}{\log{\frac{1.012}{0.990}}}\] \[n \geq 535\]

Step 3: Use the binomial probability formula

We will use the binomial probability formula to calculate the probability of the stock price increasing at least \(n = 535\) times: \[P(X \geq n) = \sum_{k=n}^{1000} \binom{1000}{k} p^k (1-p)^{1000-k}\] Where \(X\) is the number of times the stock price increases out of 1000.

Step 4: Compute the probability

Insert the values of \(n\), \(p\), and the binomial coefficient: \[P(X \geq 535) = \sum_{k=535}^{1000} \binom{1000}{k} (0.52)^k (1-0.52)^{1000-k}\] Calculate the sum to find the probability that the stock price will be up at least 30% after the next 1000 periods: \[P(X \geq 535) \approx 0.9795\] Approximately, the probability that the stock's price will be up at least 30 percent after the next 1000 periods is \(97.95\%\).

Key Concepts

Binomial Probability Distribution

Understanding the binomial probability distribution is crucial when dealing with scenarios involving two possible outcomes, such as predicting stock price movements. This type of distribution applies to situations with a fixed number of independent trials, each trial resulting in a 'success' or 'failure', with a consistent probability of success across all trials.

In our stock price example, each period is a trial with only two possible outcomes: the stock price goes up with a probability of p, or down with a probability of 1-p. To compute the probability of the stock price increasing by at least 30% after 1000 periods, we use the binomial distribution formula.

The formula for the binomial probability of exactly k successes in n trials is given by:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
where \(P(X = k)\) is the probability of k increases, n is the number of trials (1000 in this case), p is the probability of one increase, and \(\binom{n}{k}\) represents the binomial coefficient. To find the probability of at least a certain number of increases, we sum the individual probabilities for all outcomes from that number to the total number of trials.

Independent Random Variables

Key to applying the binomial distribution model to stock price changes is the assumption that successive movements are independent random variables. This means the outcome of one period does not influence or change the outcomes of others. In practical terms, if the stock price goes up in one period, this has no bearing on whether it will go up or down in the next period.

For our exercise, each period's movement is considered an independent event with its own outcome of either a price increase, with probability p, or a decrease. Because these are independent, we can apply the multiplicative rule of probability to combine individual probabilities across multiple periods. This independence is crucial because if the price movements were dependent, then the past movements could influence the likelihood of future changes, and we would not be able to use the binomial probability distribution as we do.

Logarithmic Inequality

The step involving logarithmic inequality is significant in our stock price movement problem as it helps us solve for the number of times the stock price must increase to meet or exceed our target gain.

When we are faced with an inequality like \((1.012)^n(0.990)^{1000-n} \geq 1.3\), it’s not straightforward to solve for n directly. However, by applying logarithms—an operation that helps us deal with exponents—we can transform the inequality into a more workable form. Logarithms convert multiplication into addition and division into subtraction, which simplifies the process of finding the value of n.

By taking the log of both sides and using the properties of logarithms, we converted the multiplicative inequality into an additive one. This allowed us to isolate n and determine that the stock price must increase at least 535 times out of 1000 in order to have a 30% increase in stock price. This step is critical for finding the threshold number of successes needed to subsequently apply the binomial probability distribution formula.