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Journal of Engineering Science and Technology Review

Are Financial Markets an Aspect of Quantum World?

Ovidiu Sorin Racorean

Published Feb 1, 2015• 1 citation
Publisher

Abstract

Writing the article "Time independent pricing of options in range bound markets" [1], the question in the title came naturally to my mind.It is stated, in the above article, that in certain market conditions the stock price is subjected to an equation that exactly matches a time independent Schrodinger equation.The time independent equation for options valuation is used further to explain a stock market phenomenon that resembles an α particle decay tunneling effect.The transmission coefficient for the stock price tunneling effect it is also deduced.Although, it may not have important impact in quantum physics, the philosophical aspects residing in the use of quantum mechanics for stock market specific are very important.

DOI
10.25103/jestr.081.07

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Writing the article "Time independent pricing of options in range bound markets" [1], the question in the title came naturally to my mind.It is stated, in the above article, that in certain market conditions the stock price is subjected to an equation that exactly matches a time independent Schrodinger equation.The time independent equation for options valuation is used further to explain a stock market phenomenon that resembles an α particle decay tunneling effect.The transmission coefficient for the stock price tunneling effect it is also deduced.Although, it may not have important impact in quantum physics, the philosophical aspects residing in the use of quantum mechanics for stock market specific are very important.
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 price of a given stock is exactly known only at the time of sale when the stock is between the traders. If we know the price (owner) then we have no information on the owner (price). A more general description including cases when we have partial information on both price and ownership is obtained by using the quantum mechanics methods. The relation price-ownership is similar to the relation position-momentum. Our approach is based on the mathematical formalism used in the case of quantum systems with finite-dimensional Hilbert space. The linear operator corresponding to the ownership is obtained from the linear operator corresponding to the price by using the finite Fourier transform. In our idealized model, the Schrodinger type equation describing the time evolution of the stock price is solved numerically.
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.
Modern approaches to stock pricing in quantitative finance are typically founded on the 'Black-Scholes model' and the underlying 'random walk hypothesis'. Empirical data indicate that this hypothesis works well in stable situations but, in abrupt transitions such as during an economical crisis, the random walk model fails and alternative descriptions are needed. For this reason, several proposals have been recently forwarded which are based on the formalism of quantum mechanics. In this paper we apply the 'SCoP formalism', elaborated to provide an operational foundation of quantum mechanics, to the stock market. We argue that a stock market is an intrinsically contextual system where agents' decisions globally influence the market system and stocks prices, determining a nonclassical behavior. More specifically, we maintain that a given stock does not generally have a definite value, e.g., a price, but its value is actualized as a consequence of the contextual interactions in the trading process. This contextual influence is responsible of the non-Kolmogorovian quantum-like behavior of the market at a statistical level. Then, we propose a 'sphere model' within our 'hidden measurement formalism' that describes a buying/selling process of a stock and shows that it is intuitively reasonable to assume that the stock has not a definite price until it is traded. This result is relevant in our opinion since it provides a theoretical support to the use of quantum models in finance.
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.
We pursue the quantum-mechanical challenge to the efficient market hypothesis for the stock market by employing the quantum Brownian motion model. We utilize the quantum Caldeira-Leggett master equation as a possible phenomenological model for the stock-market-prices fluctuations while introducing the external harmonic field for the Brownian particle. Two quantum regimes are of particular interest: the exact regime as well as the approximate regime of the pure decoherence ("recoilless") limit of the Caldeira-Leggett equation. By calculating the standard deviation and the kurtosis for the particle's position observable, we can detect deviations of the quantum-mechanical behavior from the classical counterpart, which bases the efficient market hypothesis. By varying the damping factor, temperature as well as the oscillator's frequency, we are able to provide interpretation of different economic scenarios and possible situations that are not normally recognized by the efficient market hypothesis. Hence we recognize the quantum Brownian oscillator as a possibly useful model for the realistic behavior of stock prices.