The Nairobi Stock Exchange (N.S.E) was founded in 1954 as a voluntary organization of the stock brokers and is now one of the most active capital markets in Africa where market players buy and sell shares and other securities. The stock prices usually vary with time and this can be attributed to factors such as economic growth, climatic changes, government policies and political atmosphere. In this book, our intention was to verify whether the price dynamics follow a random walk process or mean reversion. This may help market players understand the dynamics of prices so that they can make meaningful decision. We use one of the tests known as of Dickey-Fuller Test for unit root in a simple regression model of prices return and the parameters are estimated by the method of Ordinary Least Squares (O.L.S) Estimates. The book provides an insight for testing historical data received during trade of securities by market players in understanding the behavior of price changes. If stock price movement contain large transitory component, then for long horizon investors, the stock market may be much less risky.