Quantum Brownian motion model for the stock market

Abstract

It is believed by the majority today that the efficient market hypothesis is imperfect because of market irrationality. Using the physical concepts and mathematical structures of quantum mechanics, we construct an econophysics framework for the stock market, based on which we analogously map massive numbers of single stocks into a reservoir consisting of many quantum harmonic oscillators and their stock index into a typical quantum open system-a quantum Brownian particle. In particular, the irrationality of stock transactions is quantitatively considered as the Planck constant within Heisenberg's uncertainty relationship of quantum mechanics in an analogous manner. We analyze real stock data of Shanghai Stock Exchange of China and investigate fat-tail phenomena and non-Markovian behaviors of the stock index with the assistance of the quantum Brownian motion model, thereby interpreting and studying the limitations of the classical Brownian motion model for the efficient market hypothesis from a new perspective of quantum open system dynamics.

Quantum spatial-periodic harmonic model for daily price-limited stock markets

“…Ifh → 0, the spatial-periodic harmonic model degenerates to a classical one; a continuous E with no energy band is implied. Hence, we recognize the phenomena studied here as the ''quantum'' phenomena of stock markets, i.e., irrational phenomena [11]. It is noted that the non-zero Gaussian distribution of the ground state in Fig.…”

“…It is noted that the non-zero Gaussian distribution of the ground state in Fig. 1(a) whenh > 0 should not be identified as the well-known fat-tail Gaussian distribution of the stock return [11]: the former one accounts for the extra fluctuation caused by irrationality (or ''quantum'' phenomena), while the latter one is related to the classical statistical behavior of the entering time-varying information; this is a basic result following the treatment of quantum harmonic oscillators as canonical ensembles in quantum statistics [16].…”

“…with H n the Hermite polynomials) [26]. The physical quantities m, ω, andh separately represent the capital, the characteristic frequency, and the uncertainty of irrational transaction of the stock [11,13]: apparently, the inertia of capital against price fluctuation is analogous to the mass m; the characteristic frequency, i.e., the inverse of the response time against excitation from outside information, is analogous to the oscillating frequency ω; and more crucially, the uncertainty (or the so-called generalized uncertainty for discrete price [27]) of irrational transaction is analogous to the Planck constanth, which acts as a link between the intrinsic probability of quantum mechanics and persistent fluctuation caused by irrationality in stock markets [11].…”

“…After the noteworthy work of linking economic research with the fundamental concepts and methods of statistical physics in the 1990s [1,2], econophysics has soon burgeoned as a new interdisciplinary area, from which quantum finance [3] is then specifically introduced for applying quantum physics to finance [4][5][6][7][8][9][10][11]. With the help of quantum mechanics, rather than the wide-known classical oscillator model for stocks introduced in Ref.…”

“…With the help of quantum mechanics, rather than the wide-known classical oscillator model for stocks introduced in Ref. [12], it is recently discovered that a single stock should be treated as a quantum harmonic oscillator [13] not only excited by external information while damping to its ground state but also holding a persistent small scale of fluctuation; a stock index should be treated as a quantum Brownian particle with ensemble of stocks as a thermal reservoir [11], and a quantum Brownian model is introduced in order to explain the fat tail phenomena [14] and long-term non-Markovian features [15] of stock indices by applying the theory of quantum open systems [16]. Combining physics models with financial tools, we are able to study the underlying physical concepts of finance and economics and handle with financial problems more effectively.…”