Nonlocal Diffusions and The Quantum Black-Scholes Equation: Modelling the Market Fear Factor

TL;DR

This paper links quantum stochastic processes with nonlocal diffusions to model market fear, introducing quantum effects into Black-Scholes equations and extending to multi-variable systems for bid-offer spreads.

Contribution

It establishes a novel connection between quantum stochastic processes and nonlocal diffusions, enabling quantum-inspired market models with a fear factor component.

Findings

Quantum Black-Scholes can be expressed in integral form for simulation.

Unitary transformations introduce market fear effects into classical models.

Extended models for bid-offer spread dynamics are developed.

Abstract

In this paper, we establish a link between quantum stochastic processes, and nonlocal diffusions. We demonstrate how the non-commutative Black-Scholes equation of Accardi & Boukas (Luigi Accardi, Andreas Boukas, 'The Quantum Black-Scholes Equation', Jun 2007, available at arXiv:0706.1300v1) can be written in integral form. This enables the application of the Monte-Carlo methods adapted to McKean stochastic differential equations (H. P. McKean, 'A class of Markov processes associated with nonlinear parabolic equations', Proc. Natl. Acad. Sci. U.S.A., 56(6):1907-1911, 1966) for the simulation of solutions. We show how unitary transformations can be applied to classical Black-Scholes systems to introduce novel quantum effects. These have a simple economic interpretation as a market `fear factor', whereby recent market turbulence causes an increase in volatility going forward, that is not linked to either the local volatility function or an additional stochastic variable. Lastly, we extend this system to 2 variables, and consider Quantum models for bid-offer spread dynamics.