Semiclassical Constant Elasticity of Variance Model for Option Pricing: An Analytical Approach

Abstract This paper is devoted to obtaining closed-form solutions for the semiclassical approximation of the heat kernel of the diffusion equation, as defined by the constant elasticity of variance (CEV) option pricing model. This model is a natural, non-trivial extension of the classical Black-Scholes-Merton model. One of the key points is that our calculations are based on the Van Vleck-Morette determinant instead of the Van Vleck determinant used by other authors, giving rise to highly complicated implicit formulas rather than the simple explicit formula we obtain. Another reason for this simplification is that we use the more powerful Hamiltonian formulation instead of the Lagrangian one. In fact, we compute the Van Vleck-Morette determinant in two different ways: first, by solving the classical Hamiltonian equations, and second, by solving the variational equations. Furthermore, our calculations reveal a necessary exponential factor in the prefactor of the kernel that was missing in previous work.