# PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black-Scholes Equation

## TL;DR

This paper explores the connection between quantum stochastic processes and nonlocal diffusions in finance, introducing a PT symmetric quantum approach to develop a non-Gaussian kernel for the quantum Black-Scholes model, capturing real market behaviors.

## Contribution

It establishes a novel link between nonlocal diffusions and quantum stochastic processes and applies PT symmetric quantum mechanics to enhance the quantum Black-Scholes framework with non-Gaussian features.

## Key findings

- Nonlocal diffusions can be represented as quantum stochastic processes.
- Path integral and PT symmetry methods produce non-Gaussian kernels.
- Market behaviors emerge naturally from the quantum model.

## Abstract

The Accardi-Boukas quantum Black-Scholes framework, provides a means by which one can apply the Hudson-Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers-Moyal expansion, and this provides useful tools to understand their behaviour. In this paper we develop further links between quantum stochastic processes, and nonlocal diffusions, by inverting the question, and showing how certain nonlocal diffusions can be written as quantum stochastic processes. We then go on to show how one can use path integral formalism, and PT symmetric quantum mechanics, to build a non-Gaussian kernel function for the Accardi-Boukas quantum Black-Scholes. Behaviours observed in the real market are a natural model output, rather than something that must be deliberately included.