š¯’«š¯’Æ Symmetry, Non-Gaussian Path Integrals, and the Quantum Blackā€“Scholes Equation

  title={š¯’«š¯’Æ Symmetry, Non-Gaussian Path Integrals, and the Quantum Blackā€“Scholes Equation},
  author={William Hicks},
The Accardiā€“Boukas quantum Blackā€“Scholes framework, provides a means by which one can apply the Hudsonā€“Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramersā€“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 certainā€¦Ā 

Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion

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 howā€¦

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

The Accardi-Boukas quantum Black-Scholes equation 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 Kolmogorovā€¦

Pseudo-Hermiticity and Removing Brownian Motion From Finance

In this article we apply the methods of quantum mechanics to the study of the financial markets. Specifically, we discuss the Pseudo-Hermiticity of the Hamiltonian operators associated to the typicalā€¦

Quantum effects in an expanded Blackā€“Scholes model

Abstract The limitations of the classical Blackā€“Scholes model are examined by comparing calculated and actual historical prices of European call options on stocks from several sectors of the S &Pā€¦

Solving the Deformed Woodsā€“Saxon Potential with $$\eta $$ Ī· -Pseudo-hermetic Generator

In this paper, we present a general method to solve the non-hermetic potentials with PT symmetry using the definition of two $$\eta $$ Ī· -pseudo-hermetic and first-order operators. This generatorā€¦

Application of the Laplace Homotopy Perturbation Method to the Blackā€“Scholes Model Based on a European Put Option with Two Assets

In this paper, the Laplace homotopy perturbation method (LHPM) is applied to obtain the approximate solution of Blackā€“Scholes partial differential equations for a European put option with two assets.ā€¦



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

  • William Hicks
  • Mathematics
    Communications on Stochastic Analysis
  • 2018
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,ā€¦

The Black-Scholes pricing formula in the quantum context.

  • W. SegalI. Segal
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1998
The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context, providing a basis for a corresponding generalization of the Ito theory of stochastic integration.

Quantum Mechanics, Path Integrals and Option Pricing: Reducing the Complexity of Finance

Quantum Finance represents the synthesis of the techniques of quantum theory (quantum mechanics and quantum field theory) to theoretical and applied finance. After a brief overview of the connectionā€¦


Motivated by the work of Segal and Segal on the Black-Scholes pricing formula in the quantum context, we study a quantum extension of the Black-Scholes equation within the context ofā€¦

Quantum Ito's formula and stochastic evolutions

Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operatorā€¦

Introduction to š¯’«š¯’Æ-symmetric quantum theory

In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probabilityā€¦

Quantum Physics: A Functional Integral Point of View

This book is addressed to one problem and to three audiences. The problem is the mathematical structure of modem physics: statistical physics, quantum mechanics, and quantum fields. The unity ofā€¦


  • H. McKean
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1966
Introduction.-The familiar connection between the Brownian motion and the differential operator f -> f"/2, based upon the fact that the Brownian transition function (27rt)-1' exp[-(b a)2/2t] is alsoā€¦

Quantum Theory for Mathematicians

1 The Experimental Origins of Quantum Mechanics.- 2 A First Approach to Classical Mechanics.- 3 A First Approach to Quantum Mechanics.- 4 The Free Schrodinger Equation.- 5 A Particle in a Squareā€¦