A quantum statistical approach to simplified stock markets

Dipartimento di Metodi e Modelli Matematici, Facoltà di Ingegneria, Università di Palermo, I - 90128 Palermo, Italy
Physica A: Statistical Mechanics and its Applications (Impact Factor: 1.73). 10/2009; 388(20):4397-4406. DOI: 10.1016/j.physa.2009.07.006
Source: RePEc


We use standard perturbation techniques originally formulated in quantum
(statistical) mechanics in the analysis of a toy model of a stock market which
is given in terms of bosonic operators. In particular we discuss the
probability of transition from a given value of the {\em portfolio} of a
certain trader to a different one. This computation can also be carried out
using some kind of {\em Feynman graphs} adapted to the present context.

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    • "In some older papers of one of us (F.B.), [6]-[9], the role of information was, in a certain sense, simply incorporated by properly choosing some of the constants defining the Hamiltonian of the system we were considering. The Hamiltonian is adopted to mimic and describe the interactions between the traders, [5]. "
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    ABSTRACT: This paper shows that Hamiltonians and operators can also be put to good use even in contexts which are not purely physics based. Consider the world of finance. The work presented here {models a two traders system with information exchange with the help of four fundamental operators: cash and share operators; a portfolio operator and an operator reflecting the loss of information. An information Hamiltonian is considered and an additional Hamiltonian is presented which reflects the dynamics of selling/buying shares between traders. An important result of the paper is that when the information Hamiltonian is zero, portfolio operators commute with the Hamiltonian and this suggests that the dynamics are really due to the information. Under the assumption that the interaction and information terms in the Hamiltonian have similar strength, a perturbation scheme is considered on the interaction parameter. Contrary to intuition, the paper shows that up to a second order in the interaction parameter, a key factor in the computation of the portfolios of traders will be the initial values of the loss of information (rather than the initial conditions on the cash and shares). Finally, the paper shows that a natural outcome from the inequality of the variation of the portfolio of trader one versus the variation of the portfolio of trader two, begs for the introduction of `good' and `bad' information. It is shown that `good' information is related to the reservoirs (where an infinite set of bosonic operators are used) which model rumors/news and external facts, whilst `bad' information is associated with a set of two modes bosonic operators.
    Physica Scripta 12/2014; 90(1). DOI:10.1088/0031-8949/90/1/015203 · 1.13 Impact Factor
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    • "allows one to use the formalism of quantum mechanics in option pricing [1] [2] [4] [7]. In the new variable the conditions ( "
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    ABSTRACT: We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials and numerically compare its solution with the solution of the Black-Scholes equation.
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    • "Stock market, as one of the most important financial instruments, plays an unshakable role in basic research of finance and economics. After the noticeable works of relating economic research with fundamental concepts and methods of statistic physics in the 90s [1] [2], econophysics soon burgeons as a new interdisciplinary area, from which quantum finance is then specifically introduced for applying quantum physics to finance [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]. With help of quantum mechanics [15], it is remarked that a single stock can be treated as a quantum harmonic oscillator, excited by external information, meanwhile damping to its ground state [6], while the stock index behaves as a quantum Brownian particle with assemble of stocks as a thermal reservoir [14], of which a quantum Brownian model is introduced in order to explain fat tail phenomena [16] and long-term non-Markovian features [17] by applying the theory of quantum open systems [18]. "
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    ABSTRACT: We investigate the behavior of stocks in daily price-limited stock markets by purposing a quantum spatial-periodic harmonic model. The stock price is presumed to oscillate and damp in a quantum spatial-periodic harmonic oscillator potential well. Complicated non-linear relations including inter-band positive correlation and intra-band negative correlation between the volatility and the trading volume of stocks are derived by considering the energy band structure of the model. The validity of price limitation is then examined and abnormal phenomena of a price-limited stock market (Shanghai Stock Exchange) of China are studied by applying our quantum model.
    Physica A: Statistical Mechanics and its Applications 05/2014; 438. DOI:10.1016/j.physa.2015.06.041 · 1.73 Impact Factor
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