On Noncommutative Quantum Mechanics and the Black-Scholes Model

TL;DR

This paper develops two new quantum mechanical representations of the Black-Scholes model using geometric quantization, and extends these to noncommutative quantum mechanics, including a two-degree-of-freedom system related to the Merton-Garman model.

Contribution

It introduces novel geometric quantization methods for the Black-Scholes model and extends them to noncommutative quantum mechanics, providing new insights into financial modeling.

Findings

Constructed two quantum representations of Black-Scholes using Laplace-Beltrami operators.

Generalized the models via noncommutative quantum mechanics.

Connected the two-degree-of-freedom system to the Merton-Garman model.

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.