## The Geometric Brownian Motion Model

Based on extensive analysis of historical data and other considerations, we generally assume in finance that prices of stocks that do not pay any dividend follow a special type of stochastic process known as geometric Brownian motion. The geometric Brownian motion model assumes or implies the following properties for stock prices:

■ They are continuous in time and value.

■ They follow a Markov process, meaning that only the current stock price is relevant for predicting future prices; in this respect, the stock's price history is irrelevant.

■ The proportional return for a stock over a very short period of time is normally distributed.

■ The price of a stock is lognormally distributed.

■ The continuously compounded return for a stock is normally distributed.

The model also implies that the longer we plan to hold a stock, the more uncertain we become about the stock's final price, that is, the more widely the actual final price may vary from the expected final price. However, the longer we plan to hold a stock, the more certain we can be about earning the expected rate of return. These may sound contradictory, but we will see why they are not.

We know that the first assumption about stock prices—that they are continuous in time and value, meaning that stock prices can be observed at all times and they change continuously—does not strictly hold. Markets are closed during nights and weekends, and stock prices can change only in steps of multiples of full cents. Nonetheless, this is a reasonable assumption and it makes modeling stock prices easier.

The second assumption—that stock prices follow a Markov process—is essentially the same as the weak form of the efficient market hypothesis, which says that the future price of a stock cannot be predicted based on its price history. For example, whether a stock's price was up or down yesterday (or a minute ago) or by how much tells us nothing about what is going to happen to it in the future. If a stock has been going up for a few days, it does not mean that it has gained some momentum and is now more likely to go up for the next few days as well.

This, of course, contradicts the claims of most Wall Street experts, especially the chartists, that stock prices can be predicted by analyzing their price history in various sophisticated ways. There is overwhelming evidence to show that these claims are erroneous with maybe a few minor exceptions. The support levels, resistance levels, momentum and so forth that are discussed with so much passion and conviction on Wall Street do not really help to predict future stock prices.

Let us now discuss the other three properties in some details.

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### Responses

• steven hernandez
Do stocks follow brownian motion?
6 years ago