Problem 57 - A First Course in Probability

Question

A simplified model for the movement of the price of a stock supposes that on each day the stock's price either moves up 1 unit with probability \(p\) or moves down 1 unit with probability \(1-p .\) The changes on different days are assumed to be independent. (a) What is the probability that after 2 days the stock will be at its original price? (b) What is the probability that after 3 days the stock's price will have increased by 1 unit? (c) Given that after 3 days the stock's price has increased by 1 unit, what is the probability that it went up on the first day?

Step-by-Step Solution

Answer

(a) The probability that after 2 days the stock will be at its original price is \(2p(1-p)\). (b) The probability that after 3 days the stock's price will have increased by 1 unit is \(3p^2(1-p)\). (c) Given that after 3 days the stock's price has increased by 1 unit, the probability that it went up on the first day is \(\frac{2}{3}\).

Step 1: (a) Probability of returning to original price after 2 days

To return to the original price after two days, the stock has to either move up and then down or move down and then up. The movements are independent, so we can calculate the probability of each scenario and then add them together. Case 1: Up, then down. Probability = \(p * (1 - p)\) Case 2: Down, then up. Probability = \((1 - p) * p\) Total probability of returning to original price after 2 days = \(p(1-p) + (1-p)p = 2p(1-p)\)

Step 2: (b) Probability of stock price increasing by 1 unit after 3 days

To have an increase of 1 unit after three days, there must be two up movements and one down movement. There are three possible orders for these movements (UUD, UDU, DUU). The probability for each scenario can be calculated and then we sum them up. Case 1: Up, Up, Down. Probability = \(p * p * (1 - p) = p^2(1-p)\) Case 2: Up, Down, Up. Probability = \(p * (1 - p) * p = p^2(1-p)\) Case 3: Down, Up, Up. Probability = \((1 - p) * p * p = p^2(1-p)\) Total probability of stock price increasing by 1 unit after 3 days = \(p^2(1-p) + p^2(1-p) + p^2(1-p) = 3p^2(1-p)\)

Step 3: (c) Probability of stock price going up on the first day given an increase of 1 unit after 3 days

We are given that the stock's price increased by 1 unit after 3 days. This can occur in cases 2 and 3 (UDU and DUU) from part (b). We want to know the probability that it went up on the first day among these cases. The conditional probability can be calculated using Bayes' theorem: \(P(\text{Up on 1st day} | \text{Increase by 1 after 3 days}) = \frac{P(\text{Increase by 1 after 3 days} | \text{Up on day 1}) * P(\text{Up on day 1})}{P(\text{Increase by 1 after 3 days})}\) We already know the probability of an increase by 1 unit after 3 days, which is \(3p^2(1-p)\) from part (b). Given that the stock went up on the first day, we can have two cases for an increase of 1 unit after 3 days: UDU and UUD. The probability of each case is: Case 1: Up, Down, Up. Probability = \(p * (1 - p) * p = p^2(1-p)\) Case 2: Up, Up, Down. Probability = \(p * p * (1 - p) = p^2(1-p)\) Now we can sum them up and substitute into the Bayes' theorem equation: \(P(\text{Up on 1st day} | \text{Increase by 1 after 3 days}) = \frac{(2p^2(1-p)) * p}{3p^2(1-p)} = \frac{2}{3}\) So, the probability of the stock price going up on the first day, given that it increased by 1 unit after 3 days, is \(\frac{2}{3}\).

Key Concepts

Understanding Conditional Probability

Conditional probability is a fundamental concept in probability theory that describes the likelihood of an event occurring given that another event has already occurred. This notion can transform an uncertain scenario into one with more information, thereby affecting the outcome's probability. It is formally defined as the probability of an event A given the occurrence of another event B and is mathematically represented by the notation P(A|B).

In the context of the stock price movement exercise, conditional probability is crucial for understanding part (c) of the problem. After observing that the stock's price increased by 1 unit after 3 days, we're asked to calculate the likelihood that the price went up on the first day. Here, we're dealing with a conditional probability because the occurrence of the event 'increase by 1 unit after 3 days' provides us with additional information to reassess the probability of the first day's movement.

Using Bayes' theorem, a powerful formula in probability, we can determine such probabilities by relating the likelihood of the 'condition' (the increase after 3 days) to the likelihood of the 'target event' we are interested in (the stock going up on the first day). It can be particularly helpful when dealing with a series of independent events, where each event does not influence the occurrence of others.

Independent Events in Probability

In probability theory, events are independent if the occurrence of one event does not affect the likelihood of the other events. This means that the occurrence of one event gives no information about whether the other event will occur; the probabilities are unaffected.

For example, consider the toss of a fair coin: whether it lands heads or tails on the first toss has no influence on what it will land on the second toss. The probability of getting heads on the first toss is the same regardless of what happened in previous tosses. Independent events are a key assumption in many statistical models as they greatly simplify the probability calculations.

In the exercise we are examining, the daily stock price movements are assumed to be independent events. This independence allows us to easily compute the probability of the stock returning to its original price after 2 days (part a) or of the stock increasing by 1 unit after 3 days (part b) by simply multiplying the individual probabilities of each day's movement.

The Stock Price Movement Model

A stock price movement model is a mathematical framework used to describe and predict the changes in stock prices over time. In the simplified model given in our exercise, the stock price on each day moves either up or down by 1 unit. The probability of the price going up on any given day is represented by 'p', while the probability of it going down is '1-p' due to the law of total probability which states that all possible outcomes must sum up to 1.

The given model assumes that the movements on different days are independent events, which means that the movement on any given day does not influence the movement on any other day. This simplifies the computation of probabilities over multiple days. In practice, such models can become very complex, often considering more variables and relying on different theories of financial economics, like the random walk hypothesis or the efficient-market hypothesis.

Understanding such models is crucial for aspiring financiers and statisticians as they provide basic foundational knowledge towards comprehending more advanced financial instruments and their respective pricing models.