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

In this paper, we establish a link between quantum stochastic processes, and non-local diffusions. We demonstrate how the non-commutative Black-Scholes equation of Accardi & Boukas (Luigi Accardi, Andreas Boukas, The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, vol.2, no.2, pp.155-170, 2006) can be written in integral form. This enables the application of the Monte-Carlo methods adapted to McKean stochastic differential equations 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