Crossing Stocks and the Positive Grassmannian I: The Geometry Behind Stock Market

2014

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A simple and elegant arrangement of stock components of a portfolio (market index-DJIA) in a recent paper [1], has led to the construction of crossing of stocks diagram. The crossing stocks method revealed hidden remarkable algebraic and geometrical aspects of stock market. The present paper continues to uncover new mathematical structures residing from crossings of stocks diagram by introducing topological properties stock market is endowed with. The crossings of stocks are categorized as overcrossings and undercrossings and interpreted as generators of braid that stocks form in the process of prices quotations in the market. Topological structure of the stock market is even richer if the closure of stocks braid is considered, such that it forms a knot. To distinguish the kind of knot that stock market forms, Alexander-Conway polynomial and the Jones polynomials are calculated for some knotted stocks. These invariants of knots are important for the future practical applications topological stock market might have. Such application may account of the relation between Jones polynomial and phase transition statistical models to provide a clear way to anticipate the transition of financial markets to the phase that leads to crisis. The resemblance between braided stocks and logic gates of topological quantum computers could quantum encode the stock market behavior.

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Braided and Knotted Stocks in the Stock Market: Anticipating the Flash Crashes

2014

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“…A series of recent papers [15], [16], [17] formally introduced and developed the braid diagrams of stocks prices time series in a portfolio and sketched their connection with the topological methods of quantum computation.…”

Section: Introductionmentioning

confidence: 99%

Quantum Gates and Quantum Circuits of Stock Portfolio

2015

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In quantum computation, series of quantum gates have to be arranged in a predefined sequence that led to a quantum circuit in order to solve a particular problem. What if the sequence of quantum gates is known but both the problem to be solved and the outcome of the so defined quantum circuit remain in the shadow? This is the situation of the stock market. The price time series of a portfolio of stocks are organized in braids that effectively simulate quantum gates in the hypothesis of Ising anyons quantum computational model. Following the prescriptions of Ising anyons model, 1-qubit quantum gates are constructed for portfolio composed of four stocks. Adding two additional stocks at the initial portfolio result in 2-qubit quantum gates and circuits. Hadamard gate, Pauli gates or controlled-Z gate are some of the elementary quantum gates that are identified in the stock market structure. Addition of other pairs of stocks, that eventually represent a market index, like Dow Jones industrial Average, it results in a sequence of n-qubit quantum gates that form a quantum code. Deciphering this mysterious quantum code of the stock market is an issue for future investigations.

Decoding Stock Market Behavior with the Topological Quantum Computer

2014

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A surprising image of the stock market arises if the price time series of all Dow Jones Industrial Average stock components are represented in one chart at once. The chart evolves into a braid representation of the stock market by taking into account only the crossing of stocks and fixing a convention defining overcrossings and undercrossings. The braid of stocks prices has a remarkable connection with the topological quantum computer. Using pairs of quasi-particles, called non-abelian anyons, having their trajectories braided in time, topological quantum computer can effectively simulate the stock market behavior encoded in the braiding of stocks. In a typically topological quantum computation process the trajectories of non-abelian anyons are manipulated according to the braiding of stocks and the outcome reflects the probability of the future state of stock market. The probability depends only on the Jones polynomial of the knot formed by plat closing the quantum computation. The Jones polynomial of the knotted stock market acts, making a parallel with the common financial literature, in a topological quantum computation as a counterpart of a classical technical indicator in trading the stock market. The type of knot stock market formed is an indicator of its future tendencies.