Semiclassical CEV Option Pricing Model: an Analytical Approach

Paper: arXiv 2411.18154

Authors: Jose A. Capitán, Jose Lope-Alba, Juan J. Morales-Ruiz

Abstract

This paper is devoted to obtain closed form solutions for the semiclassical (or WKB) approximation of the heat kernel propagator of the diffusion equation defined by the constant elasticity variance (CEV) option pricing 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. In fact, we compute this determinant in two different ways: by means of the solution of the classical Hamiltonian equations, and by solving the variational equations. Furthermore, the calculation reveals an exponential factor in the prefactor of the kernel not considered in previous works.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous analytical framework applying semiclassical (WKB) approximations to the CEV option pricing model via Hamiltonian mechanics and Differential Galois theory. While highly advanced mathematically with clear corrections to previous literature, it is entirely theoretical and lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute WKB Heat Kernel<br>Propagator for CEV Option Model] --> B[Inputs: CEV Diffusion Equation &<br>Classical Hamiltonian System]
    
    B --> C[Methodology: Apply Semiclassical/WKB<br>Approximation Approach]
    
    C --> D[Compute Van Vleck-Morette Determinant]
    
    D --> E1[Path 1: Solve Classical<br>Hamiltonian Equations]
    D --> E2[Path 2: Solve Variational<br>Jacobi Equations]
    
    E1 --> F[Derive Closed-Form Solution<br>for Kernel Prefactor]
    E2 --> F
    
    F --> G[Key Finding: Discovered Missing<br>Exponential Factor in Kernel Prefactor]