Pseudo-Hermiticity and Removing Brownian Motion From Finance

@article{Hicks2020PseudoHermiticityAR,
  title={Pseudo-Hermiticity and Removing Brownian Motion From Finance},
  author={William Hicks},
  journal={arXiv: Mathematical Finance},
  year={2020}
}
  • William Hicks
  • Published 1 September 2020
  • Mathematics
  • arXiv: Mathematical 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 typical partial differential equations of Mathematical Finance (such as the Black-Scholes equation) and how this relates to the non-arbitrage condition. We propose that one can use a Schrodinger equation to derive the probabilistic behaviour of the financial market, and discuss the benefits of doing so… 
View PDF on arXiv
1 Citations

Figures from this paper

One Citation

Citation Type

Citation Type

Wild Randomness and the Application of Hyperbolic Diffusion in Financial Modelling
The application of the Cauchy distribution has often been discussed as a potential model of the financial markets. In particular the way in which single extreme, or "Black Swan", events can impact…

References

SHOWING 1-10 OF 30 REFERENCES
The Black-Scholes pricing formula in the quantum context.
  • W. Segal, I. Segal
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1998
TLDR
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.
𝒫𝒯 Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation
TLDR
It is shown how certain nonlocal diffusions can be written as quantum stochastic processes, and 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.
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,…
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 Kolmogorov…
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 operator…
A path integral approach to option pricing with stochastic volatility : Some exact results
The Black-Scholes formula for pricing options on stocks and other securities has been generalized by Merton and Garman to the case when stock volatility is stochastic. The derivation of the price of…
A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit
A discussion on embedding the Black-Scholes option pricing model in a quantum physics setting
A note on Wick products and the fractional Black-Scholes model
TLDR
It is pointed out that the definition of the self-financing trading strategies and/or thedefinition of the value of a portfolio used in the above papers does not have a reasonable economic interpretation, and thus that the results in these papers are not economically meaningful.
THE QUANTUM BLACK-SCHOLES EQUATION
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 of…
...
1
2
3
...