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arXiv:1905.07257 · Communications on Stochastic Analysis

A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

Will Hicks

Published Jan 1, 2019• 8 citations

Abstract

The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

DOI: 10.2139/ssrn.3384829

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The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

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.

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 (cf [1]) can be written in integral form.This enables the application of the Monte-Carlo methods adapted to McKean stochastic differential equations (cf [18]) 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.

In this article we model a financial derivative price as an observable on the market state function. We apply geometric techniques to integrating the Heisenberg Equation of Motion. We illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return. The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus. In particular we aim to differentiate randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is fundamental to a quantum system (Heisenberg Equation of Motion).

In this article we model a financial derivative price as an observable on a market state function, with a view to understanding how some of the non-commutative behaviour of the financial market impacts the dynamics.We integrate the Heisenberg Equation of Motion, by using a Riemannian metric, and illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return.The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework ([1]) which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus ([15]).In particular we aim to differentiate between randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is intrinsic to a quantum system (Heisenberg Equation of Motion).