On Noncommutative Quantum Mechanics and the Black-Scholes Model

Paper: arXiv 2502.00938

Authors: Abraham Espinoza-García, Pablo Vega-Lara, Luis Rey Díaz-Barrón, F. Teodoro Hernández Grovas

Abstract

Two novel and direct quantum mechanical representations of the Black-Scholes model are constructed based on the (Wick-rotated) quantization of two specific mechanical systems. The quantum setup is achieved by means of the associated Laplace-Beltrami operator (one for each model), and not by merely applying the usual naive rule. Additionally, the clear identification of the geometric content of the underlying classical framework is exploited in order to arrive at a noncommutative quantum mechanics generalization of the Black-Scholes model. We also consider a system consisting of two degrees of freedom whose (Wick-rotated) quantization leads to a model which can be seen as related to the Merton-Garman family. This model is also generalized via noncommutative quantum mechanics.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical physics approach to the Black-Scholes model using differential geometry, Laplace-Beltrami operators, and noncommutative quantum mechanics. However, it is entirely mathematical and lacks empirical data, backtesting, or financial market validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Construct Quantized & Noncommutative Black-Scholes Models] --> B[Input: Classical Mechanical Systems & BS / Merton-Garman Equations]
    B --> C[Method: Apply Wick Rotation]
    C --> D[Compute Quantum Operators via Laplace-Beltrami Operator]
    D --> E[Introduce Noncommutative Quantum Mechanics Framework]
    E --> F[Outcomes: Rigorous Geometric BS & Merton-Garman Noncommutative Models]