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Journal of Applied Mathematics and Physics

On the Interconnectedness of Schrodinger and Black-Scholes Equation

Ognjen Vukovic

Published Jan 1, 2015• 17 citations
Publisher

Abstract

The following paper tries to derive a Black-Scholes equation by using tools of quantum physics pertaining in that sense to Hamiltonian operator, path integrals, completeness equation, introducing ket and bra vectors. Schrodinger Hamiltonian is presented and compared to Black-Scholes-Schrodinger Hamiltonian. Similarity was demonstrated and it was proved that Schrodinger Hamiltonian was Hermitian while Black-Scholes Hamiltonian was anti-Hermitian. By using Schrodinger equation, price of option was implemented in the Schrodinger equation and by using Black-Scholes Hamiltonian. Black-Scholes equation was derived and a new and really powerful approach was demonstrated that could have immense application in the quantitative analysis and asset pricing.

DOI
10.4236/jamp.2015.39137

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The following paper tries to derive a Black-Scholes equation by using tools of quantum physics pertaining in that sense to Hamiltonian operator, path integrals, completeness equation, introducing ket and bra vectors. Schrodinger Hamiltonian is presented and compared to Black-Scholes-Schrodinger Hamiltonian. Similarity was demonstrated and it was proved that Schrodinger Hamiltonian was Hermitian while Black-Scholes Hamiltonian was anti-Hermitian. By using Schrodinger equation, price of option was implemented in the Schrodinger equation and by using Black-Scholes Hamiltonian. Black-Scholes equation was derived and a new and really powerful approach was demonstrated that could have immense application in the quantitative analysis and asset pricing.
Options are financial derivatives on an underlying security. The Schrodinger and\nHeisenberg approach to the quantum mechanics together with the Dirac matrix\napproaches are applied to derive the Black-Scholes formula and the quantum Cox-\nRubinstein formula.\nThe quantum mechanics approach to option pricing is based on the interpretation\nof the option price as the Schrodinger wave function of a certain quantum mechanics\nmodel determined by Hamiltonian H. We apply this approach to continuous\ntime market models generated by Levy processes.\nIn the discrete time formulization, we construct both self-adjoint and non selfadjoint\nquantum market. Moreover, we apply the discrete time formulization\nand analyse the quantum version of the Cox-Ross-Rubinstein Binomial Model.\nWe find the limit of the N-period bond market, which convergences to planar\nBrownian motion and then we made an application to option pricing in planar\nBrownian motion compared with Levy models by Fourier techniques and Monte\nCarlo method.\nFurthermore, we analyse the quantum conditional option price and compare for\nthe conditional option pricing in the quantum formulization. Additionally, we\nestablish the limit of the spectral measures proving the convergence to the geometric\nBrownian motion model. Finally, we found Binomial Model formula and\nPath integral formulization gave are close to the Black-Scholes formula.
Following system is investigated with its corresponding properties in detail: A Quantum Model Of Option Pricing: When Black–Scholes Meets Schrödinger And Its Semi-Classical Limit, Black–Scholes Option Pricing Model In A More General Quantum Physics Setting, Perfect Market Equilibrium State Postulated By The Black–Scholes Model Represents A Particular Case., Controlled Markov Processes And Mathematical Finance, Controlled Markov Processes And Mathematical Finance, Dynamic Optimization And Its Relation To Classical And Quantum Constrained Systems, Consistency Relation, Inter-Occurrence Times And Universal Laws In Finance, Earthquakes And Genomes, Quantum Cognition Beyond Hilbert Space Ii, Quantum Master Equation For Long-Term Prognosis.. The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-654991-661307.pdf
This paper deals with European share option pricing using quantum physics methods. These contingent claims are usually priced using the Black-Scholes equation. This nonlinear parabolic equation is based on geometric Brownian motion model of the stock price stochastic process. Similar processes also appear among quantum particles and are described by the time-dependent Schrödinger equation. In this paper, the option pricing based on the Schrödinger equation approach is proposed. Using Wick transformation, the Black-Scholes equation is transformed into the equivalent Schrödinger equation. The Fourier separation method is used to find analytical solutions to this equation. The last square method is used to calibrate the Schrödinger model based on real market data. Numerical results are provided and discussed.
The Schrödinger equation describes how quantum states evolve according to the Hamiltonian of the system. For physical systems, we have it that the Hamiltonian must be a Hermitian operator to ensure unitary dynamics. For anti-Hermitian Hamiltonians, the Schrödinger equation instead models the evolution of quantum states in imaginary time. This process of imaginary time evolution has been used successfully to calculate the ground state of a quantum system. Although imaginary time evolution is non-unitary, the normalised dynamics of this evolution can be simulated on a quantum computer using the quantum imaginary time evolution (QITE) algorithm. In this paper, we broaden the scope of QITE by removing its restriction to anti-Hermitian Hamiltonians, which allows us to solve any partial differential equation (PDE) that is equivalent to the Schrödinger equation with an arbitrary, non-Hermitian Hamiltonian. An example of such a PDE is the famous Black-Scholes equation that models the price of financial derivatives. We will demonstrate how our generalised QITE methodology offers a feasible approach for real-world applications by using it to price various European option contracts modelled according to the Black-Scholes equation.
In this chapter the concept of the Hamiltonian is introduced in the pricing of options. Hamiltonians occur naturally in finance; to demonstrate this the analysis of the Black–Scholes equation is recast in the formalism of quantum mechanics. It is then shown how the Hamiltonian plays a central role in the general theory of option pricing. The Hamiltonian formulation provides new tools for obtaining solutions for option pricing; two key concepts related to the Hamiltonian are (a) eigenfunctions and (b) potentials. Knowledge of all the eigenfunctions of a Hamiltonian yields an exact solution for a large class of path-dependent and path-independent options. For example, barrier options can be modelled by placing constraints on the eigenfunctions of the Hamiltonian. The potentials are a means for defining new financial instruments, and for modelling path-dependent options. Essentials of quantum mechanics It is shown in this chapter that option pricing in finance has a mathematical description that is identical to a quantum system; hence the key features of quantum theory are briefly reviewed. Quantum theory is a vast subject that forms the bedrock of contemporary physics, chemistry and biology. Only those aspects of quantum mechanics are reviewed that are relevant for the analysis of option pricing. In classical mechanics the position of a particle at time t , denoted by x t , is a deterministic function of t , and is given by Newton's law of motion. Classical mechanics is analogous to the case of the evolution of a stock price with zero volatility (σ = 0) that yields a deterministic evolution of the stock price.