Solutions in quadratures to the CEV model

Abstract

Cox introduced the constant elasticity of variance (CEV) model in 1975, in order to capture this inverse relationship between the stock price and its volatility. An important parameter in the model is the parameter β, the elasticity of volatility. We use Kovacic's algorithm to derive, for all half-integer values of β, all solutions 'in quadratures' of the CEV ordinary differential equation (CEV ODE). These solutions give rise, by separation of variables, to simple solutions to the CEV partial differential equation. In particular, when β = …, …5⁄2, …2, …3⁄2, …1, 1, 3⁄2, 2, 5⁄2, …, we obtain four classes of denumerably infinite elementary function solutions, when β = …½ and β = ½ we obtain two classes of denumerably infinite elementary function solutions, whereas, when β = 0 we find two elementary function solutions.