# Random walk theory

Random walk theory |
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**Random walk theory** is a financial market theory that states that stock prices change randomly and follow no particular direction. This theory suggests that the past movement or direction of a stock price or market cannot be used to predict its future movement. It suggests that stock prices have an equal probability of increasing or decreasing in the short term.

- The random walk theory assumes that stock prices are determined by a variety of factors, such as news, company earnings, and investor sentiment, which all move randomly and independently of one another.
- The random walk theory implies that stock prices have no memory and that past price movements have no bearing on future price movements.
- The random walk theory also suggests that it is impossible to outperform the market in the long run, as all stocks will eventually reach their fair value, regardless of how they are traded.

## Example of Random walk theory

A classic example of the random walk theory can be seen by observing the stock prices of a company over a period of time. For example, if a company's stock price was $50 at the beginning of the week, the random walk theory suggests that the stock price could either increase or decrease to an unknown level by the end of the week. *The stock price could increase to $55, decrease to $45, or remain at $50. *The random walk theory states that all of these outcomes have the same probability, which is impossible to predict. *The stock price could also increase or decrease by different amounts, such as increasing by $1 or decreasing by $2, or any other amount. *This demonstrates that stock prices move randomly, and that past price movements have no bearing on future price movements.

## Formula of Random walk theory

Random walk theory states that stock prices change randomly and follow no particular direction. This theory can be explained mathematically using the following formula\[\begin{equation} P (t) = P_0 + \sum_{i=1}^t X_i \end{equation}\]

Where P (t) is the price at time t, P_{0} is the initial price, and X_{i} is the random change in price at each time step. This formula suggests that stock prices have no memory and that past price movements have no bearing on future price movements. It implies that stock prices are determined by a variety of factors, such as news, company earnings, and investor sentiment, which all move randomly and independently of one another. In conclusion, the random walk theory implies that it is impossible to outperform the market in the long run, as all stocks will eventually reach their fair value, regardless of how they are traded.

## When to use Random walk theory

Random walk theory can be used to evaluate and strategize investments. It can be used to determine the risk and return of a portfolio, and to develop strategies for reducing risk and improving returns. Additionally, it can be used to evaluate the performance of a portfolio over time, and to determine whether a portfolio is performing better or worse than the market as a whole. Random walk theory can also be used to analyze the predictability of stock prices, and to determine the degree to which stock prices are influenced by economic or political events. In conclusion, random walk theory can be used to evaluate, strategize, and analyze investments, and to determine the predictability of stock prices.

## Types of Random walk theory

There are three main types of random walk theory: weak, semi-strong, and strong.

- The weak form of the random walk theory states that past prices of a stock do not affect future prices, but that information contained in the price itself can be used to make predictions.
- The semi-strong form of the random walk theory states that all publicly available information is reflected in the current stock price and that it cannot be used to predict future prices.
- The strong form of the random walk theory states that all information, including private information, is already reflected in the current price and cannot be used to predict future prices. In conclusion, there are three main types of random walk theory, each of which suggests that it is impossible to predict future stock prices.

## Steps of Random walk theory

The steps of the Random Walk Theory include:

- First, the theory assumes that stock prices are determined by a variety of factors, such as news, company earnings, and investor sentiment, which all move randomly and independently of one another.
- Second, the theory suggests that past price movements have no bearing on future price movements.
- Third, the theory implies that it is impossible to outperform the market in the long run, as all stocks will eventually reach their fair value, regardless of how they are traded.
- Finally, the theory can be explained mathematically using the formula mentioned above. In summary, the Random Walk Theory suggests that stock prices change randomly and independently of one another, and that it is impossible to predict future price movements.

## Advantages of Random walk theory

Random walk theory has several advantages.

- First, it helps investors to understand market movements, as it suggests that stock prices move randomly and independently of one another.
- Second, it suggests that even though it is impossible to predict future stock prices, it is possible to achieve returns in line with the market by investing in a diversified portfolio.
- Third, it provides a framework for understanding the behavior of stock prices and other financial markets.
- Fourth, it allows investors to understand why it is difficult to outperform the market in the long run.

## Limitations of Random walk theory

Random walk theory has several limitations.

- First, it assumes that all investors have perfect knowledge of the stock market, which is rarely the case.
- Second, it assumes that prices move randomly and independently of one another, which is also not always true.
- Third, it assumes that past price movements do not affect future price movements, which is not always the case.
- Finally, it assumes that the stock market is efficient, meaning that all available information is priced into a stock, which is also not always the case. All of these assumptions limit the validity of the random walk theory and make it difficult to apply in real-world situations.

The random walk theory is closely related to the efficient market hypothesis, which states that all information about a security is already reflected in the price of the security. This hypothesis suggests that it is impossible to consistently generate above-average returns by trading securities, as all information is already priced into the market. Another related theory is the weak form of the efficient market hypothesis, which suggests that past performance of securities is not a reliable predictor of future performance.

In conclusion, the random walk theory is related to several other financial market theories, such as the efficient market hypothesis and the weak form of the efficient market hypothesis. These theories suggest that it is impossible to consistently generate above-average returns by trading securities or to predict future price movements.

## Suggested literature

- Banasiak, J., & Mika, J. R. (1998).
*Singularly perturbed telegraph equations with applications in the random walk theory*. Journal of Applied Mathematics and Stochastic Analysis, 11(1), 9-28.