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

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

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 u...

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Main Author: Will Hicks
Format: Article
Language:English
Published: MDPI AG 2019-01-01
Series:Entropy
Subjects:
quantum Black–Scholes
non-Gaussian kernels
quantum stochastic calculus
PT symmetric quantum mechanics
Online Access:https://www.mdpi.com/1099-4300/21/2/105
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spelling doaj-0d5e7556144543b2915a2d99f45fa4ca2020-11-24T21:06:00ZengMDPI AGEntropy1099-43002019-01-0121210510.3390/e21020105e21020105PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes EquationWill Hicks0Investec Bank PLC, 30 Gresham Street, London EC2V 7QP, UKThe 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 <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">PT</mi> </semantics> </math> </inline-formula> 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.https://www.mdpi.com/1099-4300/21/2/105quantum Black–Scholesnon-Gaussian kernelsquantum stochastic calculusPT symmetric quantum mechanics
collection DOAJ
language English
format Article
sources DOAJ
author Will Hicks
spellingShingle Will Hicks
PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
Entropy
quantum Black–Scholes
non-Gaussian kernels
quantum stochastic calculus
PT symmetric quantum mechanics
author_facet Will Hicks
author_sort Will Hicks
title PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
title_short PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
title_full PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
title_fullStr PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
title_full_unstemmed PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
title_sort pt symmetry, non-gaussian path integrals, and the quantum black–scholes equation
publisher MDPI AG
series Entropy
issn 1099-4300
publishDate 2019-01-01
description 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 <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">PT</mi> </semantics> </math> </inline-formula> 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.
topic quantum Black–Scholes
non-Gaussian kernels
quantum stochastic calculus
PT symmetric quantum mechanics
url https://www.mdpi.com/1099-4300/21/2/105
work_keys_str_mv AT willhicks ptsymmetrynongaussianpathintegralsandthequantumblackscholesequation
_version_ 1716767052291637248

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