Results on the CEV Process, Past and Present

Abstract

We consider the Constant Elasticity of Variance (CEV) process, carefully re- visiting the relationships between its transition density and that of the non-central chi-squared distribution, and establish a symmetry principle which readily explains many classical results. The principle also sheds light on the cases in which the CEV parameter exceeds one and the forward price process is a strictly local martingale. An analysis of this parameter regime shows that the widely-quoted formula for the price of a plain vanilla European call option requires a correction term to achieve an arbitrage free price. We discuss Monte Carlo simulation of the CEV process, the specifics of which depend on the parameter regime, and compare the results against the analytic expressions for plain vanilla European option prices. We find good agreement. Using these techniques, we also verify that the expected forward price is a strictly local martingale when the CEV parameter is greater than one.

Keywords: CEV Process, Bessel Process, Local Martingale

JEL Classification: C20, C63