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.
It seems to be very unlikely that all relevant information in the stock market could be fully encoded in a geometrical shape. Still, the present paper will reveal the geometry behind the stock market transactions. The prices of market index (DJIA) stock components are arranged in ascending order from the smallest one in the left to the highest in the right. In such arrangement, as stock prices changes due to daily market quotations, it could be noticed that the price of a certain stock get over /under the price of a neighbor stock. These stocks are crossing. Arranged this way, the diagram of successive stock crossings is nothing else than a permutation diagram. From this point on the financial and combinatorial concepts are netted together to build a bridge connecting the stock market to a beautiful geometrical object that will be called stock market polytope. The stock market polytope is associated with the remarkable structure of positive Grassmannian .This procedure makes all the relevant information about the stock market encoded in the geometrical shape of the stock market polytope more readable.
Assuming that price of the underlying stock is moving in range bound, the Black-Scholes formula for options pricing supports a separation of variables. The resulting time-independent equation is solved employing different behavior of the option price function and three significant results are deduced. The first is the probability of stock price penetration through support or resistance level, called transmission coefficient. The second is the distance that price will go through once stock price penetrates out of the range bound. The last one is a predicted short time dramatic fall in the stock volatility right ahead of price tunneling. All three results are useful tools that give market practitioners valuable insights in choosing the right time to get involved in an option contract, about how far the price will go in case of a breakout, and how to correctly interpret volatility downfalls. JEL Classification: G13, G17.
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.
We investigate the emergence of thermodynamic arrow of time in the context of AdS/CFT correspondence. We show that, on the CFT side, if the two copies of the field theory are not initially correlated the entropy can only increase such that a definite orientation of the thermodynamic arrow of time is imposed. Conversely, in a high-correlation environment, the entropy can either increase or decrease, such that there is no opportunity for the dominance of one direction of time over the other. On the gravity side, we construct the structure of geometric dual by considering the notion of spacetime sidedness and time-orientability. Accordingly, we conjecture that the entanglement of the CFTs in the thermofield double state, impose the connection of the two sides of the spacetime forming a one sided spacetime. In addition, disentangling the degrees of freedom of the two CFTs, results in disconnecting the two sides of the spacetime. In essence, the maximal entanglement between the two copies of CFT builds, in the geometric dual, a connection between the two sides of spacetime.
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