In this thesis, we present an extended version of the statistical properties of the maximum Sharpe ratio based on the article by Maller et al (2005). The Sharpe ratio is a basic tool in finance, especially for performance evaluation. Its use as a summary measure of the (excess) return achieved by a portfolio, as compared with the risk taken, is established among theorists and practitioners. The maximum Sharpe ratio plays a fundamental role in the portfolio optimization problem, in which it can be used to locate an optimal point on the efficient frontier (the tangent point) which determines the constitution of the portfolio of risky assets which, in equilibrium, will be held in some proportion by any investor. From the statistical point of view, the study of the Sharpe ratio helps investors to make correct decisions under different circumstances. Note that, in practice, we use the estimated Sharpe ratio which may cause statistical error. The structure of this thesis is as follows. After reviewing primary concepts of the portfolio theory in chapter one, we present some statistical concepts and theorems in chapter two. Chapter three contains an introduction to the Sharpe ratio, the maximum Sharpe ratio and some of their properties. Some other performance measures are also discussed in this chapter. Chapter four is the most important part of this thesis, in which we try to study the statistical properties of the maximum Sharpe ratio. In chapter four, we study bias, consistency, asymptotic distribution (under two different conditions) and the value at risk property of the maximum Sharpe ratio, respectively. In chapter five, the portfolio selection problem is studied when returns follow a stable distribution. Finally, simulations and the result of our investigations is presented in chapter six.