We generalize the recently proposed quantum model for the stock market by Zhang and Huang to make it consistent with the discrete nature of the stock price. In this formalism, the price of the stock and its trend satisfy the generalized uncertainty relation and the corresponding generalized Hamiltonian contains an additional term proportional to the fourth power of the trend. We study a driven infinite quantum well where information as the external field periodically fluctuates and show that the presence of the minimal trading value of stocks results in a positive shift in the characteristic frequencies of the quantum system. The connection between the information frequency and the transition probabilities is discussed finally.
Econophysics as an interdisciplinary research field was started in mid 1990s by physicists who are interested to apply theories and models originally developed in physics for solving the complex problems appeared in economics, specially in financial markets [1]. Because of the stochastic nature of the financial markets, the majority of tools for market analysis such as stochastic processes and nonlinear dynamics have their roots in statistical physics. Besides statistical physics, other branches of physics and mathematics have a major role in the development of econophysics. The sophisticated tools developed in quantum mechanics such as perturbation theory, path integral (Feynman-Kac) methods, random matrix and the spin-glass theories are shown to be useful for option pricing and portfolio optimization.
Among theoretical physics, quantum field theory has a special role to reveal the intricacies of nature from quantum electrodynamics to critical phenomena. For instance, it can be used to model portfolios as a financial field and describes the change of financial markets via path integrals and differential manifolds [2,3].
The application of quantum mechanics to financial markets has attracted much attention in recent years to model the finance behavior with the laws of quantum mechanics and it is becoming now a rather established fact [4][5][6][7][8][9][10]. For instance, Schaden, contrary to stochastic descriptions, used the quantum theory to model secondary financial markets to show the importance of trading in determining the value of an asset [11]. He considered securities and cash held by investors as the wave function to construct the Hilbert space of the stock market. Another useful application of quantum theory to trading strategies is quantum game theory which is the generalization of classical game theory to the quantum domain [12,13]. This theory is primarily based on quantum cryptography and contains superimposed initial wave functions, quantum entanglement of initial wave functions, and superposition of strategies in addition to its classical counterpart.
At this point, it is worth explaining why quantum mechanics is essential to study the behavior of the stock market. Classical mechanics which is described by Newton’s law of motion is deterministic in the sense that it exactly predicts the position of a particle at each instant of time. This is similar to the evolution of a stock price with zero volatility (σ = 0) that results in a deterministic evolution of the stock price. However, in the context of quantum mechanics, the evolution of the position of the particle has a probabilistic interpretation which is similar to the evolution of a stock price with a non-zero volatility (σ = 0) [3]. Note that there is a close connection between the Black-Scholes-Merton (BSM) equation [14,15] and the Schrödinger equation: The position of a quantum particle is a random variable in quantum mechanics, and similarly, the price of a security is a random variable in finance. Also, the Schrödinger equation admits a complex wave function, whereas the BSM equation is a real partial differential equation which can be considered as the Schrödinger equation for imaginary time. Haven showed that BSM equation is a special case of the Schrödinger equation where markets are assumed to be efficient [16]. Indeed, various mathematical structures of quantum theory such as probability theory, state space, operators, Hamiltonians, commutation relations, path integrals, quantized fields, fermions and bosons have natural and useful applications in finance. In the language of Schaden, “The evolution into a superposition of financial states and their measurement by transaction is my understanding of quantum finance” [17].
Recently, Zhang and Huang have proposed a new quantum financial model in econophysics and defined wave functions and operators of the stock market to construct the Schrödinger equation for studying the dynamics of the stock price [18]. They solved the corresponding partial differential equation of a given Hamiltonian to find a quantitative description for the volatility of the Chinese stock market.
In their formalism, the wave function ψ(℘, t) is considered as the price distribution, where ℘ denotes the stock price and t is the time. There, the stock price is approximately considered as a continuousvariable. However, the stock price is actually a discrete variable and admits a non-zero minimal price length (∆℘) min = 0 which depends on the stock market’s local currency. In this paper, we incorporate the fact of discrete nature of the stock price with the quantum description of the stock market. We show that the uncertainty relation between the price and its trend and the form of the Hamiltonian should be modified to make the quantum formulation consistent with discrete property of the stock price. Note that, Bagarello has also tried to present quantum financial models which describe quantities which assume di
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