A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions in Option Pricing, Review, and Integration

Abstract

One of the important issues in option pricing is to find a stock return distribution that allows the stock rate of return and its volatility to depend on each other. Cox’s (Notes on option pricing I: constant elasticity of diffusions, unpublished draft, Stanford University, 1975) Constant Elasticity of Variance (CEV) diffusion generates a family of distributions for such a purpose. The main goal of this paper is to review and show the procedures of how such process and its resulting option pricing formula are derived. First, we show how the density function of the CEV diffusion is identified and we demonstrate the option formula by using the Cox and Ross (Journal of Financial Economics 145–166, 1976) methodology. Then, we transform the solution into a non-central chi-square distribution. Finally, a number of approximation formulas are provided.