The constant elasticity of variance option pricing model

Abstract

The Constant Elasticity of Variance (CEV) Model was first presented in 1976 by John Cox and Stephen Ross as an extension to the famous Black-Scholes European Call Option Pricing Model of 1972/3. Unlike the Black-Scholes model, which is accessible to anyone with a pocket calculator and tables of the standard normal distribution, the CEV solution consists of a pair of infinite summations of gamma density and survivor functions. Its derivation rested on the risk-neutral pricing theory and the results of Feller. Moreover, descriptions of this model in journal and text-book literature frequently contained errors. One difficulty in implementation of the Black-Scholes model is that one of its arguments is an unobserved parameter σ, the share price volatility. Much research has concentrated on estimating this parameter with a general conclusion that it is better to imply this volatility from observed option prices, than to estimate it from stock price data. In the case of the CEV model there are two unobserved parameters, δ2, with a relationship to the Black-Scholes parameter, and β, which defines the relationship between share price level and the variance of the instantaneous rate of return on the share. Attempts made to estimate this second parameter in the early 1980's were not altogether satisfactory, perhaps condemning the CEV model to obscurity. A breakthrough was made in 1989 with a paper by Mark Schroder, who devised a method of evaluating the CEV option prices using the non-central Chi-Squared probability distribution, hence facilitating significantly simpler computation of the prices for those with suitable statistical software I use the statistical package SPLUS extensively in this thesis. This thesis attempts to summarise the development of the CEV model, with comparisons made to the industry standard, the Black-Scholes model. The elegance of Schroder's method is also made clear. Joint estimation of the two parameters of the CEV model, δ and β, is attempted using both simulated data, and a sample of stocks traded on the Australian Stock Exchange. In this section, it appears that significant improvements can be made to earlier estimation methods. Finally, I would like to note that this thesis is primarily a statistical analysis of a financial topic. As a consequence of this, the focus of my analysis differs to that of articles in the financial journal literature, and that of finance texts. Furthermore, the literature on the CEV model is sparse, and in general theorems therein are stated without proof. Some theorems found in this thesis reflect the statistical nature of the analysis and are hence absent from the financial literature which I have surveyed and referenced. Throughout the thesis, I have attempted to make it clear when an idea or proof follows previously established material. Unattributed material is generally that which I have worked on with my supervisors' guidance, but which is not found in the references I have used.