Asymptotics of Barrier Option Pricing Under the CEV Process

Abstract

We apply a singular perturbation analysis to some option pricing models. To illustrate the technique we first consider the European put option under the standard Black–Scholes model, with or without barriers. Then we consider the same option under the constant elasticity of variance (CEV) assumption, which is also called the square root process. In the CEV model the variability effects in the evolution of the asset, on which the option is based, are proportional to the square root of the asset value. We also consider the CEV model with barriers, and this leads to a rich asymptotic structure. The analysis assumes that the variability is small and employs the ray method of geometrical optics and matched asymptotic expansions.